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| In [[mathematics]], more specifically [[ring theory]], a branch of [[abstract algebra]], the '''Jacobson radical''' of a [[Ring (mathematics)|ring]] ''R'' is an [[ideal (ring theory)|ideal]] which consists of those elements in ''R'' which [[Annihilator (ring theory)|annihilate]] all [[Simple module|simple]] right ''R''-[[Module (mathematics)|modules]]. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left-right symmetric. The Jacobson radical of a ring is frequently denoted by ''J''(''R'') or rad(''R''); however to avoid confusion with other [[radical of a ring|radicals of rings]], the former notation will be preferred in this article. The Jacobson radical is named after [[Nathan Jacobson]], who was the first to study it for arbitrary rings in {{harv|Jacobson|1945}}.
| | == 物理的なスリルを == |
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| The Jacobson radical of a ring has numerous internal characterizations, including a few definitions which successfully extend the notion to rings without [[Multiplicative identity|unity]]. The [[radical of a module]] extends the definition of the Jacobson radical to include modules. The Jacobson radical plays a prominent role in many ring and module theoretic results, such as [[Nakayama's lemma]].
| | それは日光に輝く、突然Duoliaoyifenかすかな光沢の上に排出シャオヤンの顔の集合体を共有するように浮き沈みの破裂、喉の集約ダウン最後のものはヒスイのような一枚のように、吐き出し、そして非常に魅力的な注意 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-10.html カシオの時計]。<br>物理的なスリルを<br>あり、他のアクションは、シャオヤンはあぐらをかいて体が空にまっすぐ立っている春の一般的のようです座っていない、ねじれたツイストボディ、崔翔が少しヒントを拡大しそう上げ、彼の口の弧の間の骨だった [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 腕時計 バンド]。<br><br>ゆっくり手から、そっとシャオヤンをこすり二本の指は、突然、非常に弱い銀色「色」稲妻の線、それが前面に指からシューという音で結構です [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 掛け時計]。<br><br>「精錬の成功それは？ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオソーラー時計] '小さな指Nalv雷を見て、シャオヤンの眉は、それを洗練風と雷を強制、タッチの外に喜びを抵抗することができず、彼の想像力はとても難しく、危険ではない、これらの優れた弟子の風と雷裁判所であり、それはこの中にある |
| <!-- For instance, if ''R'' is a ring, ''J''(''R'') equals the intersection of all ''maximal right ideals'' in ''R''.<ref>Isaacs, Corollary 13.3, p. 180</ref> Somewhat remarkable is that this also equals the intersection of all ''maximal left ideals'' of ''R''.{{sfn|Isaacs|year=1993|loc=p. 182}} Although the Jacobson radical is indeed an ideal, this is not entirely obvious from the previous two characterizations and hence other characterizations are preferred.<ref>Isaacs, p. 180</ref> Despite the nature of these characterizations, the intersection of all ''maximal (double-sided) ideals'' in ''R'' need not equal ''J''(''R'') – for instance, when ''R'' is a the [[endomorphism ring]] of a [[vector space]] with [[countable]] [[dimension of a vector space|dimension]] over a field ''F'', it is known that ''R'' has precisely three ideals, {0},''I'' and ''R'', however since ''R'' is [[von Neumann regular]] J(''R'')=0. {{harv|Lam|2001|loc=Ex. 3.15|p=46}}-->
| | 相关的主题文章： |
| <!-- A computationally convenient notion when working with the Jacobson radical of a ring, is the notion of [[Quasiregular element|quasiregularity]].<ref>Isaacs, p. 180</ref> In particular, every element of a ring's Jacobson radical is quasiregular, and the Jacobson radical can be characterized as the unique right ideal of a ring, maximal with respect to the property that each element is [[Quasiregular element|right quasiregular]].<ref>Isaacs, Theorem 13.4, p. 180</ref>{{sfn|Isaacs|year=1993|loc=p. 181}} It is not necessarily true, however, that every quasiregular element belongs to a ring's Jacobson radical.{{sfn|Isaacs|year=1993|loc=p. 181}} The notion of quasiregularity proves to be very useful in various situations discussed later{{sfn|Isaacs|year=1993|loc=p. 181}}<ref>Isaacs, Theorem 13.11, p. 183</ref> --> | | <ul> |
| <!--The Jacobson radical of a ring is also useful in studying [[Module (mathematics)|modules]] over the ring.{{sfn|Isaacs|year=1993|loc=p. 182}}<ref>Isaacs, Theorem 13.11, p. 183</ref> For instance, if ''U'' is a right ''R''-module, and ''V'' is a maximal submodule of ''U'', then ''U''·''J''(''R'') is contained in ''V'', where ''U''·''J''(''R'') denotes all products of elements of ''J''(''R'') (the "scalars") with elements in ''U'', on the right.{{sfn|Isaacs|year=1993|loc=p. 182}} Another instance of the usefulness of ''J''(''R'') when studying right ''R''-modules, is [[Nakayama's lemma]].<ref>Isaacs, Corollary 13.12, p. 183</ref>--> | | |
| <!-- In this case, the ring may not even contain a (proper) ''maximal'' right or left ideal (although, it may well contain non-trivial proper (one-sided) ideals). Thus, all of the above characterizations fail (including the characterization involving [[Quasiregular element|quasiregularity]] for this requires that the ring have unity). This problem, as well as the solution, is discussed later in the article, where the Jacobon radical is defined for rings without unity. --> | | <li>[http://www.midland-design.co.uk/cgi-bin/index.cgi http://www.midland-design.co.uk/cgi-bin/index.cgi]</li> |
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| | <li>[http://www.zixunwang.cn/home.php?mod=space&uid=28197 http://www.zixunwang.cn/home.php?mod=space&uid=28197]</li> |
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| | <li>[http://www.5231.net/home.php?mod=space&uid=491429 http://www.5231.net/home.php?mod=space&uid=491429]</li> |
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| | </ul> |
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| ==Intuitive discussion== | | == この弟子、古い '医学'、本当に静かに多くのことをやった == |
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| As with other [[radical of a ring|radicals of rings]], the '''Jacobson radical''' can be thought of as a collection of "bad" elements. In this case the "bad" property is that these elements annihilate all simple left and right modules of the ring. For purposes of comparison, consider the [[Nilradical of a ring|nilradical]] of a [[commutative ring]], which consists of all elements which are [[Nilpotent element|nilpotent]]. In fact for any ring, the nilpotent elements in the [[center (algebra)|center]] of the ring are also in the Jacobson radical.{{sfn|Isaacs|year=1993|loc=p. 181}} So, for commutative rings, the nilradical is contained in the Jacobson radical.
| | 言った： [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html 腕時計 メンズ casio] '、Danleiそうでない場合は、「色」残念ながら、この菩提ダンだけ挑発'薬 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html 電波腕時計 カシオ] 'の力が強くなりますが、あなたの体は菩提の心を持って、プラス菩提ダンのヘルプは、成功することができ、半神聖に躍進。 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html gps 腕時計 カシオ] '<br>古い古い顔が笑顔で覆われていることを「医学」を見て<br>、シャオヤンの心が多少ブロックされ、前者は二週間姿を消したそのうちのオリジナルは、この菩提ダンが出て洗練彼を助けるための場所を探して外出することであり、この、彼は今、成功率洗練されたダン「医学」のレベルの種類があまりにも低い数字ですが、彼も理解しているという「薬」古い容量、精錬菩提ダンは、十分な9「色」Danleiを誘致するためには、唯一の彼の手に骨がある場合、しかし、これが冷たい火のその後の精神、、あまり炎効果の一つである、9「カラー」Danlei描いた、行うことは非常に簡単です [http://www.nnyagdev.org/sitemap.xml http://www.nnyagdev.org/sitemap.xml]。<br>彼に<br>この弟子、古い [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html casio 腕時計 ゴールド] '医学'、本当に静かに多くのことをやった。<br><br>「先生、ありがとうございました。 ' |
| | 相关的主题文章： |
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| | <li>[http://www.putizi.net/bbs/home.php?mod=space&uid=46637 http://www.putizi.net/bbs/home.php?mod=space&uid=46637]</li> |
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| The Jacobson radical is very similar to the nilradical in an intuitive sense. A weaker notion of being bad, weaker than being a [[zero divisor]], is being a non-unit (not invertible under multiplication). The Jacobson radical of a ring consists of elements which satisfy a stronger property than being merely a non-unit – in some sense, a member of the Jacobson radical must not "act as a unit" in ''any'' [[Module (mathematics)|module]] "internal to the ring." More precisely, a member of the Jacobson radical must project under the [[Quotient map|canonical homomorphism]] to the zero of every "right division ring" (each non-zero element of which has a [[right inverse]]) internal to the ring in question. Concisely, it must belong to every maximal right ideal of the ring. These notions are of course imprecise, but at least explain why the nilradical of a commutative ring is contained in the ring's Jacobson radical.
| | == 」は、異なる火の三種類 == |
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| In yet a simpler way, we may think of the Jacobson radical of a ring as method to "mod out bad elements" of the ring – that is, members of the Jacobson radical act as 0 in the [[quotient ring]], ''R''/''J''(''R''). If ''N'' is the nilradical of commutative ring ''R'', then the quotient ring ''R''/''N'' has no nilpotent elements. Similarly for any ring ''R'', the quotient ring has ''J''(''R''/''J''(''R''))={0} and so all of the "bad" elements in the Jacobson radical have been removed by modding out ''J''(''R''). Elements of the Jacobson radical and nilradical can be therefore seen as generalizations of 0.
| | あなたの体の強さに私の魂を捨てレイ攻撃が長すぎることができないので、私はすぐに解決！ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ 掛け時計] '<br><br>Skyfireのりっぱなリマインダの声が突然シャオヤンの心に聞こえた [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ スタンダード 腕時計]。<br><br>ウェンヤン、シャオヤンで少しうなずいた彼の頭には、火災時の額とインドでの軽く指は、すぐに、奇妙な白い炎森 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ ソーラー 腕時計] '色'をフック屈筋深呼吸を取り、それが急激に跳び。<br><br>骨冷たい火はリン、シャオヤンヤシの把握、グリーンスプリットや裸火、心臓の炎症が登場し、火明かりQinglianに分類されます [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-13.html 時計 カシオ]。<br><br>シャオヤン、3フレーム間の手のひらのグリップの最後の目の前に浮かぶ三つの異なる火災の体は、突然、デイトン時間を凝集力、暴力的なテロ力の高まり、広がって航海した [http://nrcil.net/sitemap.xml http://nrcil.net/sitemap.xml]。<br><br>」は、異なる火の三種類？どのようにそれができるか？ '<br><br>シャオヤンの3火炎面を参照してください、それははるかに気分張メイ長老トリオ振る持っていないようですが、それは顔である |
| | 相关的主题文章： |
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| | <li>[http://2233777.cc/forum.php?mod=viewthread&tid=27212&extra= http://2233777.cc/forum.php?mod=viewthread&tid=27212&extra=]</li> |
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| | <li>[http://90v-birds.com/tl-r/cgi/aska-in/aska.cgi http://90v-birds.com/tl-r/cgi/aska-in/aska.cgi]</li> |
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| ==Equivalent characterizations== | | == 、シャオヤンはやや無力なため息を思わ == |
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| The Jacobson radical of a ring has various internal and external characterizations. The following equivalences appear in many noncommutative algebra texts such as {{harv|Anderson|1992|loc=§15}}, {{harv|Isaacs|1993|loc=§13B}}, and {{harv|Lam|2001|loc=Ch 2}}.
| | 「3人の若いマスターが、何が起こったのか？ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html casio 腕時計 説明書] 'シャオヤンは、彼の後ろに来て、すぐに功盛笑い、尋ねた、嘉Lieao、襲っペイン一見。<br><br>シャオヤンが顔片頭痛 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html カシオ腕時計 g-shock] '色'醜い嘉Lieaoを見て、微笑んで、何気なく言った： [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ 腕時計 激安] '嘉Lieaoマスター、シティスクエアが、私xiaojiaサイト、あなたがここで手をしたいですか？'<br><br>嘉Lieaoペインの目は冷笑シャオヤンに転じ、その後、いくつかの恐怖を見て： '？したばかりの家族の影響に依存しないでください、あなたは男であれば...」<br><br>「私が男だった場合は、私が言いたい、あなたが公正なテストに来てください、それはないですか？ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 腕時計 バンド] 'シャオヤンは突然嘉Lieao語を、振っ微笑んで中断。<br><br>嘉Lieaoの冷笑、挑発的と言った： [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ 腕時計 gps] 'はい、あなたは勇気ができますか？'<br>挑発嘉Lieaoの顔を見<br>、シャオヤンはやや無力なため息を思わ |
| | 相关的主题文章： |
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| | <li>[http://wshear.dip.jp/ws/wsbbs/bbs.cgi http://wshear.dip.jp/ws/wsbbs/bbs.cgi]</li> |
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| | <li>[http://maillists.freead.com.au/cgi-bin/mojo.cgi http://maillists.freead.com.au/cgi-bin/mojo.cgi]</li> |
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| | <li>[http://www.imc-e.net/thread-1312060-1-1.html http://www.imc-e.net/thread-1312060-1-1.html]</li> |
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| The following are equivalent characterizations of the Jacobson radical in rings with unity (characterizations for rings without unity are given immediately afterward):
| | == Nalan甘い == |
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| * ''J''(''R'') equals the intersection of all [[maximal ideal|maximal right ideals]] of the ring. It is also true that ''J''(''R'') equals the intersection of all maximal left ideals within the ring.{{sfn|Isaacs|year=1993|loc=p. 182}} These characterizations are internal to the ring, since one only needs to find the maximal right ideals of the ring. For example, if a ring is [[Local ring|local]], and has a unique maximal ''right ideal'', then this unique maximal right ideal is an ideal because it is exactly ''J''(''R''). Maximal ideals are in a sense easier to look for than annihilators of modules. This characterization is deficient, however, because it does not prove useful when working computationally with ''J''(''R''). The left-right symmetry of these two definitions is remarkable and has various interesting consequences.<ref>Isaacs, Problem 12.5, p. 173</ref>{{sfn|Isaacs|year=1993|loc=p. 182}} This symmetry stands in contrast to the lack of symmetry in the socles of ''R'', for it may happen that soc(''R''<sub>''R''</sub>) is not equal to soc(<sub>''R''</sub>''R''). If ''R'' is a non-commutative ring, ''J''(''R'') is not necessarily equal to the intersection of all maximal ''two-sided'' ideals of ''R''. For instance, if ''V'' is a countable direct sum of copies of a field ''k'' and ''R=End(V)'' (the ring of endomorphisms of ''V'' as a ''k''-module), then ''J''(''R'')=0 because ''R'' is known to be [[von Neumann regular]], but there is exactly one maximal double-sided ideal in ''R'' consisting of endomorphisms with finite-dimensional image. {{harv|Lam|2001|loc=Ex. 3.15|p=46}}
| | 、手を振って重い足は、体が急に離れて激しくNalan甘い衝突に対して、影になった。<br><br>'はそれを戦う [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html カシオ 時計 電波]！Nalan甘い！三年！'三年間の抑制、低轟音間の影の衝突を、助けることが自己喉の中で、アウト渡すことができませんでした。<br>一般的には怒っWarcraftのような正方形の中のすべての人々の目を下に<br>影は、光がスパークと長い深い痕跡を、直接、ブルーストーン上に沿って、地上源獣の足に取り付けられた [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html casio 腕時計 メンズ]。<br><br>Nalan甘い顔はまっすぐに影から冷静に見えた、彼女の風力発電法は属 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html 腕時計 メンズ casio] 'セックス'の一部であるため、好調スピードと敏捷性は、今後のシャオヤンで彼女の最も良いものであるすぐに風 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html カシオ 腕時計 ソーラー] 'スイング'風と共に去りぬメートルの範囲、Nalan甘い最終的にいくつかのアクション、つま先が床をタップし、一般的には葉の風のように体が、あっという間に、点滅し、その暴力的な紛争彼の全身が来ることです合格黒 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 腕時計 説明書] '色'の影。 |
| | | 相关的主题文章： |
| * ''J''(''R'') equals the sum of all [[superfluous submodule|superfluous right ideals]] (or symmetrically, the sum of all superfluous left ideals) of ''R''. Comparing this with the previous definition, the sum of superfluous right ideals equals the intersection of maximal right ideals. This phenomenon is reflected dually for the right socle of ''R'': soc(''R''<sub>''R''</sub>) is both the sum of [[minimal ideal|minimal right ideal]]s and the intersection of [[essential extension|essential right ideals]]. In fact, these two astounding relationships hold for the radicals and socles of modules in general.
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| * As defined in the introduction, ''J''(''R'') equals the intersection of all [[Annihilator (ring theory)|annihilators]] of [[simple module|simple]] right ''R''-modules, however it is also true that it is the intersection of annihilators of simple left modules. An ideal that is the annihilator of a simple module is known as a [[primitive ideal]], and so a reformulation of this states that the Jacobson radical is the intersection of all primitive ideals. Although this characterization is not useful computationally, or as useful as the previous two characterizations in aiding intuition, it is useful in studying modules over rings. For instance, if ''U'' is right ''R''-module, and ''V'' is a [[maximal submodule]] of ''U'', ''U''·''J''(''R'') is contained in ''V'', where ''U''·''J''(''R'') denotes all products of elements of ''J''(''R'') (the "scalars") with elements in ''U'', on the right. This follows from the fact that the [[quotient module]], ''U''/''V'' is simple and hence annihilated by ''J''(''R''). As another example, this result motivates [[Nakayama's lemma]].
| | <li>[http://www5a.biglobe.ne.jp/~charlie1/charliehoney2.cgi http://www5a.biglobe.ne.jp/~charlie1/charliehoney2.cgi]</li> |
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| * ''J''(''R'') is the unique right ideal of ''R'' maximal with the property that every element is [[Quasiregular element|right quasiregular]].<ref>Isaacs, Corollary 13.4, p. 180</ref>{{sfn|Isaacs|year=1993|loc=p. 181}} Alternatively, one could replace "right" with "left" in the previous sentence.{{sfn|Isaacs|year=1993|loc=p. 182}} This characterization of the Jacobson radical is useful both computationally and in aiding intuition. Furthermore, this characterization is useful in studying modules over a ring. [[Nakayama's lemma]] is perhaps the most well-known instance of this. Although every element of the ''J''(''R'') is necessarily [[Quasiregular element|quasiregular]], not every quasiregular element is necessarily a member of ''J''(''R'').{{sfn|Isaacs|year=1993|loc=p. 181}}
| | <li>[http://112.124.41.0/bbs/home.php?mod=space&uid=93616 http://112.124.41.0/bbs/home.php?mod=space&uid=93616]</li> |
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| * While not every quasiregular element is in ''J''(''R''), it can be shown that ''y'' is in ''J''(''R'') if and only if ''xy'' is left quasiregular for all ''x'' in ''R''. {{harv|Lam|2001|p=50}}
| | <li>[http://www.seabear.com.hk/forum/read.php?tid=17663 http://www.seabear.com.hk/forum/read.php?tid=17663]</li> |
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| For rings without unity it is possible for ''R''=''J''(''R''), however the equation that ''J''(''R''/''J''(''R''))={0} still holds. The following are equivalent characterizations of ''J''(''R'') for rings without unity appear in {{harv|Lam|2001|p=63}}:
| | </ul> |
| * The notion of left quasiregularity can be generalized in the following way. Call an element ''a'' in ''R'' left ''generalized quasiregular'' if there exists ''c'' in ''R'' such that ''c''+''a''-''ca''= 0. Then ''J''(''R'') consists of every element ''a'' for which ''ra'' is left generalized quasiregular for all ''r'' in ''R''. It can be checked that this definition coincides with the previous quasiregular definition for rings with unity.
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| * For a ring without unity, the definition of a left [[simple module]] ''M'' is amended by adding the condition that ''R•M'' ≠ 0. With this understanding, ''J''(''R'') may be defined as the intersection of all annihilators of simple left ''R'' modules, or just ''R'' if there are no simple left ''R'' modules. Rings without unity with no simple modules do exist, in which case ''R''=''J''(''R''), and the ring is called a '''radical ring'''. By using the generalized quasiregular characterization of the radical, it is clear that if one finds a ring with ''J''(''R'') nonzero, then ''J''(''R'') is a radical ring when considered as a ring without unity.
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| ==Examples==
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| * Rings for which ''J''(''R'') is {0} are called [[semiprimitive ring]]s, or sometimes "Jacobson semisimple rings". The Jacobson radical of any [[field (mathematics)|field]], any [[von Neumann regular ring]] and any left or right [[primitive ring]] is {0}. The Jacobson radical of the [[integer]]s is {0}.
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| * The Jacobson radical of the ring '''Z'''/12'''Z''' (see [[modular arithmetic]]) is 6'''Z'''/12'''Z''', which is the intersection of the maximal ideals 2'''Z'''/12'''Z''' and 3'''Z'''/12'''Z'''.
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| * If ''K'' is a field and ''R'' is the ring of all upper triangular ''n''-by-''n'' matrices with entries in ''K'', then J(''R'') consists of all upper triangular matrices with zeros on the main diagonal.
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| * If ''K'' is a field and ''R'' = ''K''<nowiki>[[</nowiki>''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub><nowiki>]]</nowiki> is a ring of [[formal power series]], then J(''R'') consists of those power series whose constant term is zero. More generally: the Jacobson radical of every [[local ring]] is the unique maximal ideal of the ring, which consists precisely of the ring's non-[[unit (algebra)|units]].
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| * Start with a finite, acyclic [[quiver (mathematics)|quiver]] Γ and a field ''K'' and consider the quiver algebra ''K''Γ (as described in the [[quiver (mathematics)|quiver article]]). The Jacobson radical of this ring is generated by all the paths in Γ of length ≥ 1.
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| * The Jacobson radical of a [[C*-algebra]] is {0}. This follows from the [[Gelfand–Naimark theorem]] and the fact for a C*-algebra, a topologically irreducible *-representation on a [[Hilbert space]] is algebraically irreducible, so that its kernel is a primitive ideal in the purely algebraic sense (see [[spectrum of a C*-algebra]]).
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| ==Properties==
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| * If ''R'' is unital and is not the trivial ring {0}, the Jacobson radical is always distinct from ''R'' since [[Maximal ideal#Properties|rings with unity always have maximal right ideals]]. However, some important theorems and conjectures in ring theory consider the case when ''J''(''R'') = ''R'' - "If ''R'' is a nil ring (that is, each of its elements is nilpotent), is the [[polynomial ring]] ''R''[''x''] equal to its Jacobson radical?" is equivalent to the open [[Köthe conjecture]]. {{harv|Smoktunowicz|2006|loc=§5|p=260}}
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| * The Jacobson radical of the ring ''R''/J(''R'') is zero. Rings with zero Jacobson radical are called [[semiprimitive ring]]s.
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| * A ring is [[semisimple algebra|semisimple]] if and only if it is [[Artinian ring|Artinian]] and its Jacobson radical is zero.
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| * If ''f'' : ''R'' → ''S'' is a [[surjective]] [[ring homomorphism]], then ''f''(J(''R'')) ⊆ J(''S'').
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| * If ''M'' is a [[finitely generated module|finitely generated]] left ''R''-[[module (mathematics)|module]] with J(''R'')''M'' = ''M'', then ''M'' = 0 ([[Nakayama's lemma]]).
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| * ''J''(''R'') contains all central nilpotent elements, but contains no [[idempotent element]]s except for 0.
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| * ''J''(''R'') contains every [[nil ideal]] of ''R''. If ''R'' is left or right [[artinian ring|Artinian]], then J(''R'') is a [[nilpotent ideal]]. This can actually be made stronger: If <math>\left\{0\right\}= T_0\subseteq T_1\subseteq \dotsb\subseteq T_k=R</math> is a [[Composition_series#For_modules|composition series]] for the right ''R''-module ''R'' (such a series is sure to exist if ''R'' is right artinian, and there is a similar left composition series if ''R'' is left artinian), then <math>\left(J\left(R\right)\right) ^k=0</math>. (Proof: Since the factors <math>T_u/T_{u-1}</math> are simple right ''R''-modules, right multiplication by any element of J(''R'') annihilates these factors. In other words, <math>\left(T_u/T_{u-1}\right)\cdot J\left(R\right)=0</math>, whence <math>T_u\cdot J\left(R\right)\subseteq T_{u-1}</math>. Consequently, induction over ''i'' shows that all nonnegative integers ''i'' and ''u'' (for which the following makes sense) satisfy <math>T_u\cdot \left(J\left(R\right)\right)^i\subseteq T_{u-i}</math>. Applying this to ''u'' = ''i'' = ''k'' yields the result.) Note, however, that in general the Jacobson radical need not consist of only the [[nilpotent]] elements of the ring.
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| * If ''R'' is commutative and finitely generated as a '''Z'''-module, then J(''R'') is equal to the [[Nilradical of a ring|nilradical]] of ''R''.
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| *The Jacobson radical of a (unital) ring is its largest superfluous right (equivalently, left) ideal.
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| *{{citation
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| |author1=Anderson, Frank W.
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| |author2=Fuller, Kent R.
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| |title=Rings and categories of modules
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| |publisher=Springer-Verlag
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| |isbn=0-387-97845-3
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| *{{citation
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| |author1=Atiyah, M. F.
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| |author2=Macdonald, I. G.
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| |title=Introduction to commutative algebra
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| |publisher=Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont.
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| |year=1969
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| |pages=ix+128 | |
| |mr=0242802 (39 #4129)}}
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| *N. Bourbaki. ''Éléments de Mathématique''.
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| *{{citation
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| |author=Herstein, I. N.
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| |author-link=Israel Nathan Herstein
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| |title=Noncommutative rings
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| |series=Carus Mathematical Monographs
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| |volume=15
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| |note=Reprint of the 1968 original;
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| With an afterword by Lance W. Small
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| |publisher=Mathematical Association of America
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| |place=Washington, DC
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| |year=1994
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| |pages=xii+202
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| |isbn=0-88385-015-X
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| |mr=1449137 (97m:16001)}}
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| * {{cite book
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| | author = Isaacs, I. M.
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| | year = 1993
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| | title = Algebra, a graduate course
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| | edition = 1st edition
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| | publisher = Brooks/Cole Publishing Company
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| | isbn = 0-534-19002-2}}
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| * {{Citation | last1=Jacobson | first1=Nathan | author1-link=Nathan Jacobson | title=The radical and semi-simplicity for arbitrary rings | doi=10.2307/2371731 | mr=12271 | year=1945 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=67 | pages=300–320}}
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| *{{citation
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| |author=Lam, T. Y.
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| |title=A first course in noncommutative rings
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| |series=Graduate Texts in Mathematics
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| |volume=131
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| |edition=2
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| |publisher=Springer-Verlag
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| |place=New York
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| |year=2001
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| |pages=xx+385
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| |isbn=0-387-95183-0
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| |mr=1838439 (2002c:16001)}}
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| *{{citation
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| |author=Pierce, Richard S.
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| |title=Associative algebras
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| |series=Graduate Texts in Mathematics
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| |volume=88
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| |note=Studies in the History of Modern Science, 9
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| |publisher=Springer-Verlag
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| |place=New York
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| |year=1982
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| |pages=xii+436
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| |isbn=0-387-90693-2
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| |mr=674652 (84c:16001)}}
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| ==See also==
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| *[[Nilradical of a ring|Nilradical]]
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| *[[Radical of a module]]
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| *[[Radical of an ideal]]
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| *[[Frattini subgroup]]
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| {{DEFAULTSORT:Jacobson Radical}}
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| [[Category:Ideals]]
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| [[Category:Ring theory]]
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