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| In [[abstract algebra]], an element ''x'' of a set with a [[binary operation]] ∗ is called an '''idempotent element''' (or just an '''idempotent''') if {{nowrap|1=''x'' ∗ ''x'' = ''x''}}. This reflects the [[idempotence]] of the binary operation on that particular element.
| | <br><br>[http://www.hostgator.com/hostgator-coupons hostgator.com]Hostgator Coupon Code - [http://thapenny.byus.net/zbxe/?document_srl=165858 http://thapenny.byus.net/zbxe/?document_srl=165858]. <br><br>I had like 17 domains hosted on single account, and never had any special troubles. First on my list is call centers. When it comes to picking the best web hosting sites to host your blog, offering unlimited web space it comes down to Fatcow and Hostgator if you want the best.The next segment is to hunt for free internet marketing strategies to put into business into your blog or webpage, present are moreover several free internet marketing tools obtainable all on the web for you to make use of. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. The best thing you can do to further your writing career is...write! There's a respectable debate as to why HostGator have won so many awards along with why they're so commonly liked by affiliate marketers along with entrepreneurs all round the earth. During the course of my aforementioned research, I decided upon NameCheap for my domain company. Click on this icon to open QuickInstall ?You will observe a lot of free software such as WordPress,b2evolution etc to install. Web hosting requirements such as bandwidth and disk space should also be taken into deliberation. If you are looking forward for hosting your Wordpress blog on hostgator this hub might be helpful in gathering information about the company before you make a decision.<br><br><br><br>The countless plugins, support and long-time use has cemented it as the go-to choice when starting a blog. The less time any webmaster spends inside the hosting account interface, the more efficient the site is. These applications and tools might be simply managed by way of a web hosting provider whose sole purpose is to make sites obtainable via the internet. There are so many different providers to choose from. The steps in Fantastico De Luxe are pretty easy to follow, but when you come to the "install in directory" box, if your entire domain will be the blog, leave this blank. These companies have made me happy, and I hope this information is helpful for those considering a new domain or hosting company. Why should you go beyond just making a website? The surprise is the fact that, HostGator has gained much more than 2,25,000 buyers inside the past 8 many years. |
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| Idempotents are especially prominent in [[ring theory]]. For general rings, elements idempotent under multiplication are tied with decompositions of modules, as well as to [[homological algebra|homological]] properties of the ring. In [[Boolean algebra]], the main objects of study are rings in which all elements are idempotent under both addition and multiplication.
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| ==Definitions==
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| An idempotent element of a ring is an element ''a'' such that {{nowrap|1=''a''<sup>2</sup> = ''a''}}.<ref>See [[Michiel Hazewinkel|Hazewinkel]] et al. (2004), p. 2.</ref> That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that {{nowrap|1=''a'' = ''a''<sup>2</sup> = ''a''<sup>3</sup> = ''a''<sup>4</sup> … = ''a''<sup>''n''</sup>}} for any positive integer ''n''.
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| There are many special types of idempotents defined after the following examples section.
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| ==Examples==
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| One may consider the ring of integers mod n, where n is [[Square-free integer|squarefree]]. By the [[Chinese Remainder Theorem]], this ring factors into the direct product of rings of integers mod ''p''. Now each of these factors is a field, so it is clear that the only idempotents will be 0 and 1. That is, each factor has 2 idempotents. So if there are ''m'' factors, there will be 2<sup>''m''</sup> idempotents.
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| We can check this for the integers mod 6, {{nowrap|1=''R'' = '''Z'''/6'''Z'''}}. Since 6 has 2 factors (2 and 3) it should have 2<sup>2</sup> idempotents.
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| : 0<sup>2</sup> = 0 = 0 (mod 6)
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| : 1<sup>2</sup> = 1 = 1 (mod 6)
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| : 2<sup>2</sup> = 4 = 4 (mod 6) | |
| : 3<sup>2</sup> = 9 = 3 (mod 6)
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| : 4<sup>2</sup> = 16 = 4 (mod 6)
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| : 5<sup>2</sup> = 25 = 1 (mod 6)
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| From these computations, 0, 1, 3, and 4 are idempotents of this ring, while 2 and 5 are not. This also demonstrates the decomposition properties described below: because {{nowrap|1=3 + 4 = 1 (mod 6)}}, there is a ring decomposition 3'''Z'''/6'''Z'''⊕4'''Z'''/6'''Z'''. In 3'''Z'''/6'''Z''' the identity is 3+6'''Z''' and in 4'''Z'''/6'''Z''' the identity is 4+6'''Z'''.
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| ;Other examples
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| There is a [[catenoid]] of idempotents in the [[coquaternion]] ring.
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| ==Types of ring idempotents==
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| A partial list of important types of idempotents includes:
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| *Two idempotents ''a'' and ''b'' are called '''orthogonal''' if {{nowrap|1=''ab'' = ''ba'' = 0}}. If ''a'' is idempotent in the ring ''R'' (with unity), then so is {{nowrap|1=''b'' = 1 − ''a''}}; moreover, ''a'' and ''b'' are orthogonal.
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| *An idempotent ''a'' in ''R'' is called a '''central idempotent''' if {{nowrap|1=''ax'' = ''xa''}} for all ''x'' in ''R''.
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| *A '''trivial idempotent''' refers to either of the elements 0 and 1, which are always idempotent.
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| *A '''primitive idempotent''' is an idempotent ''a'' such that ''aR'' is [[directly indecomposable]].
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| *A '''local idempotent''' is an idempotent ''a'' such that ''aRa'' is a [[local ring]]. This implies that ''aR'' is directly indecomposable, so local idempotents are also primitive.
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| *A '''right irreducible idempotent''' is an idempotent ''a'' for which ''aR'' is a simple module. By [[Schur's lemma]], {{nowrap|1=End<sub>''R''</sub>(''aR'') = ''aRa''}} is a division ring, and hence is a local ring, so right (and left) irreducible idempotents are local.
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| *A '''centrally primitive''' idempotent is a central idempotent ''a'' that cannot be written as the sum of two nonzero orthogonal central idempotents.
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| *An idempotent {{nowrap|1=''e'' + ''I''}} in the quotient ring ''R''/''I'' is said to '''lift modulo ''I''''' if there is an idempotent ''f'' in ''R'' such that {{nowrap|1=''f'' + ''I'' = ''e'' + ''I''}}.
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| *An idempotent of ''e'' of ''R'' is called a '''full idempotent''' if ''ReR''=''R''.
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| *A '''separability idempotent'''; see [[separable algebra]].
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| Any non-trivial idempotent ''a'' is a [[zero divisor]] (because {{nowrap|1=''ab'' = 0}} with neither ''a'' nor ''b'' being zero, where {{nowrap|1=''b'' = 1 − ''a''}}). This shows that [[integral domain]]s and [[division ring]]s don't have such idempotents. [[Local ring]]s also don't have such idempotents, but for a different reason. The only idempotent contained in the [[Jacobson radical]] of a ring is 0.
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| ==Rings characterized by idempotents==
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| *A ring is [[semisimple ring|semisimple]] if and only if every right (or every left) ideal is generated by an idempotent.
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| *A ring is [[von Neumann regular ring|von Neumann regular]] if and only if every [[finitely generated module|finitely generated]] right (or every finitely generated left) ideal is generated by an idempotent.
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| *A ring in which ''all'' elements are idempotent is called a [[Boolean ring]]. In such a ring, multiplication is commutative and every element is its own [[additive inverse]].
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| *A ring for which the [[annihilator (ring theory)|annihilator]] <math>\mathrm{r.Ann}(S)</math> every subset ''S'' of ''R'' is generated by an idempotent is called a [[Baer ring]]. If the condition only holds for all [[singleton (mathematics)|singleton]] subsets of ''R'', then the ring is a right [[Rickart ring]]. Both of these types of rings are interesting even when they lack a multiplicative identity.
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| *A ring in which all idempotents are [[Center (algebra)|central]] is called an '''Abelian ring'''. Such rings need not be commutative.
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| *A ring is [[irreducible ring|directly irreducible]] if and only if 0 and 1 are the only central idempotents.
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| *A ring ''R'' can be written as {{nowrap|1=''e''<sub>1</sub>''R'' ⊕ ''e''<sub>2</sub>''R'' ⊕ ... ⊕ ''e''<sub>n</sub>''R''}} with each ''e''<sub>i</sub> a local idempotent if and only if ''R'' is a [[semiperfect ring]].
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| *A ring is called an '''[[SBI ring]]''' or '''Lift/rad''' ring if all idempotents of ''R'' lift modulo the [[Jacobson radical]].
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| *A ring satisfies the [[ascending chain condition]] on right direct summands if and only if the ring satisfies the [[descending chain condition]] on left direct summands if and only if every set of pairwise orthogonal idempotents is finite.
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| *If ''a'' is idempotent in the ring ''R'', then ''aRa'' is again a ring, with multiplicative identity ''a''. The ring ''aRa'' is often referred to as a '''corner ring''' of ''R''. The corner ring arises naturally since the ring of endomorphisms {{nowrap|1=End<sub>''R''</sub>(''aR'') ≅ ''aRa''}}.
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| ==Role in decompositions==
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| The idempotents of ''R'' have an important connection to decompositon of ''R'' [[Module (mathematics)|modules]]. If ''M'' is an ''R'' module and {{nowrap|1=''E'' = End<sub>''R''</sub>(''M'')}} is its [[ring of endomorphisms]], then {{nowrap|1=''A'' ⊕ ''B'' = ''M''}} if and only if there is a unique idempotent ''e'' in ''E'' such that {{nowrap|1=''A'' = ''e''(''M'')}} and {{nowrap|1=''B'' = (1 − ''e'') (''M'')}}. Clearly then, ''M'' is directly indecomposable if and only if 0 and 1 are the only idempotents in ''E''.{{sfn|Anderson|Fuller|1992|loc=p.69-72}} | |
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| In the case when {{nowrap|1=''M'' = ''R''}} the endomorphism ring {{nowrap|1=End<sub>''R''</sub>(''R'') = ''R''}}, where each endomorphism arises as left multiplication by a fixed ring element. With this modification of notation, {{nowrap|1=''A'' ⊕ ''B'' = ''R''}} as right modules if and only if there exists a unique idempotent ''e'' such that {{nowrap|1=''eR'' = ''A''}} and {{nowrap|1=(1 − ''e'')''R'' = ''B''}}. Thus every module direct summand of ''R'' is generated by an idempotent.
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| If ''a'' is a central idempotent, then the corner ring {{nowrap|1=''aRa'' = ''Ra''}} is a ring with multiplicative identity ''a''. Just as idempotents determine the direct decompositions of ''R'' as a module, the central idempotents of ''R'' determine the decompositions of ''R'' as a [[Direct sum of modules|direct sum]] of rings. If ''R'' is the direct sum of the rings ''R''<sub>1</sub>,...,''R''<sub>''n''</sub>, then the identity elements of the rings ''R''<sub>''i''</sub> are central idempotents in ''R'', pairwise orthogonal, and their sum is 1. Conversely, given central idempotents ''a''<sub>1</sub>,...,''a''<sub>''n''</sub> in ''R'' that are pairwise orthogonal and have sum 1, then ''R'' is the direct sum of the rings ''Ra''<sub>1</sub>,…,''Ra''<sub>''n''</sub>. So in particular, every central idempotent ''a'' in ''R'' gives rise to a decomposition of ''R'' as a direct sum of the corner rings ''aRa'' and {{nowrap|1=(1 − ''a'')''R''(1 − ''a'')}}. As a result, a ring ''R'' is directly indecomposable as a ring if and only if the identity 1 is centrally primitive.
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| Working inductively, one can attempt to decompose 1 into a sum of centrally primitive elements. If 1 is centrally primitive, we are done. If not, it is a sum of central orthogonal idempotents, which in turn are primitive or sums of more central idempotents, and so on. The problem that may occur is that this may continue without end, producing an infinite family of central orthogonal idempotents. The condition "''R does not contain infinite sets of central orthogonal idempotents''" is a type of finiteness condition on the ring. It can be achieved in many ways, such as requiring the ring to be right [[Noetherian ring|Noetherian]]. If a decomposition {{nowrap|1=''R'' = ''c''<sub>1</sub>''R'' ⊕ ''c''<sub>2</sub>''R'' ⊕ ... ⊕ ''c''<sub>n</sub>''R''}} exists with each ''c''<sub>i</sub> a centrally primitive idempotent, then ''R'' is a direct sum of the corner rings ''c''<sub>i</sub>''Rc''<sub>i</sub>, each of which is ring irreducible.{{sfn|Lam|2001|loc=p.326}}
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| ==Relation with involutions==
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| If ''a'' is an idempotent of the endomorphism ring End<sub>''R''</sub>(''M''), then the endomorphism {{nowrap|1=''f'' = 1 − 2''a''}} is an ''R'' module [[Involution (mathematics)|involution]] of ''M''. That is, ''f'' is an ''R'' homomorphism such that ''f'' <sup>2</sup> is the identity endomorphism of ''M''.
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| An idempotent element ''a'' of ''R'' and its associated involution ''f'' gives rise to two involutions of the module ''R'', depending on viewing ''R'' as a left or right module. If ''r'' represents an arbitrary element of ''R'', ''f'' can be viewed as a right ''R''-homomorphism {{nowrap|1= ''r'' ↦ ''fr''}} so that {{nowrap|1=''ffr'' = ''r''}}, or ''f'' can also be viewed as a left ''R'' module homomorphism {{nowrap|1=''r'' ↦ ''rf''}}, where {{nowrap|1=''rff'' = ''r''}}.
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| This process can be reversed if 2 is an [[invertible element]] of ''R'':<ref>Rings in which 2 is not invertible are not hard to find. The element 2 is not invertible in any Boolean algebra, nor in any ring of [[characteristic (algebra)|characteristic]] 2.</ref> if ''b'' is an involution, then {{nowrap|1=2<sup>−1</sup>(1 − b)}} and {{nowrap|1=2<sup>−1</sup>(1 + b)}} are orthogonal idempotents, corresponding to ''a'' and {{nowrap|1=1 − ''a''}}. Thus for a ring in which 2 is invertible, the idempotent elements [[bijection|correspond]] to involutions in a one-to-one manner.
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| ==Category of ''R'' modules==
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| Lifting idempotents also has major consequences for the category of ''R'' modules. All idempotents lift modulo ''I'' if and only if every ''R'' direct summand of ''R''/''I'' has a [[projective cover]] as an ''R'' module.{{sfn|Anderson|Fuller|1992|loc=p.302}} Idempotents always lift modulo [[nil ideal]]s and rings for which ''R''/''I'' is [[completion (ring theory)#Krull topology|I-adically complete]].
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| Lifting is most important when {{nowrap|1=''I'' = J(''R'')}}, the [[Jacobson radical]] of ''R''. Yet another characterization of semiperfect rings is that they are [[semilocal ring]]s whose idempotents lift modulo J(''R'').{{sfn|Lam|2001|loc=p.336}}
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| ==Lattice of ideals==
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| One may define a [[partial order]] on the idempotents of a ring as follows: if ''a'' and ''b'' are idempotents, we write {{nowrap|1=''a'' ≤ ''b''}} [[if and only if]] {{nowrap|1=''ab'' = ''ba'' = ''a''}}. With respect to this order, 0 is the smallest and 1 the largest idempotent. For orthogonal idempotents ''a'' and ''b'', {{nowrap|1=''a'' + ''b''}} is also idempotent, and we have {{nowrap|1=''a'' ≤ ''a'' + ''b''}} and {{nowrap|1=''b'' ≤ ''a'' + ''b''}}. The [[atom (order theory)|atom]]s of this partial order are precisely the primitive idempotents. {{harv|Lam|2001|p=323}}
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| When the above partial order is restricted to the central idempotents of ''R'', a lattice structure can be given. For two central idempotents ''e'' and ''f'' the [[Boolean algebra#Operations|complement]] {{nowrap|1=¬''e'' = 1 − ''e''}} and the [[join and meet]] are given by
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| :''e'' ∨ ''f'' = ''e'' + ''f'' − ''ef''
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| and
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| :''e'' ∧ ''f'' = ''ef''.
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| The ordering now becomes simply {{nowrap|1=''e'' ≤ ''f''}} if and only if {{nowrap|1=''eR'' ⊆ ''fR''}}, and the join and meet satisfy {{nowrap|1=(''e''∨''f'')''R'' = ''eR'' + ''fR''}} and {{nowrap|1=(''e''∧''f'')''R'' = ''eR'' ∩ ''fR'' = (''eR'')(''fR'')}}. It is shown in {{harv|Goodearl|1991|p=99}} that if ''R'' is [[von Neumann regular]] and right [[injective module#Self-injective rings|self-injective]], then the lattice is a [[complete lattice]]. | |
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| == Notes ==
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| <references/>
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| == References ==
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| {{refbegin}}
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| * “[http://foldoc.org/idempotent idempotent]” at [[FOLDOC]]
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| *{{citation
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| |author=Goodearl, K. R.
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| |title=von Neumann regular rings
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| |edition=2
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| |publisher=Robert E. Krieger Publishing Co. Inc.
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| |place=Malabar, FL
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| |year=1991
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| |pages=xviii+412
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| |isbn=0-89464-632-X
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| |mr=1150975 (93m:16006)}}
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| *{{citation
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| |author1=Hazewinkel, Michiel
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| |author2=Gubareni, Nadiya
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| |author3=Kirichenko, V. V.
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| |title=Algebras, rings and modules. Vol. 1
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| |series=Mathematics and its Applications
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| |volume=575
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| |publisher=Kluwer Academic Publishers
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| |place=Dordrecht
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| |year=2004
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| |pages=xii+380
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| |isbn=1-4020-2690-0
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| |mr=2106764 (2006a:16001)}}
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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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| * {{Lang Algebra|edition=3}} p. 443
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| * Peirce, Benjamin.. [http://www.math.harvard.edu/history/peirce_algebra/index.html ''Linear Associative Algebra''] 1870.
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| *{{citation
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| |author1=Polcino Milies, César
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| |author2=Sehgal, Sudarshan K.
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| |title=An introduction to group rings
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| |series=Algebras and Applications
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| |volume=1
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| |publisher=Kluwer Academic Publishers
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| |place=Dordrecht
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| |year=2002
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| |pages=xii+371
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| |isbn=1-4020-0238-6
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| |mr=1896125 (2003b:16026)}}
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| {{refend}}
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| [[Category:Abstract algebra]]
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