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[[File:A portion of the lattice of ideals of Z illustrating prime, semiprime and primary ideals.png|A portion of the lattice of ideals of Z illustrating prime, semiprime and primary ideals|thumb|right|320px|A [[Hasse diagram]] of a portion of the lattice of ideals of the integers {{math|'''Z'''}}. The purple and red nodes indicate semiprime ideals. The purple nodes are [[prime ideal]]s, and the purple and blue nodes are [[primary ideal]]s.]]
In [[ring theory]], a branch of mathematics, '''semiprime [[ideal (ring theory)|ideal]]s''' and '''semiprime [[ring (mathematics)|ring]]s''' are generalizations of [[prime ideal]]s and [[prime ring]]s. In [[commutative algebra]], semiprime ideals are also called '''[[radical ideal]]s'''.

For example, in the ring of [[integers]], the semiprime ideals are the zero ideal, along with those ideals of the form <math>n\mathbb Z</math> where ''n'' is a [[square-free integer]]. So, <math>30\mathbb Z</math> is a semiprime ideal of the integers, but <math>12\mathbb Z\,</math> is not.

The class of semiprime rings includes [[semiprimitive ring]]s, [[prime ring]]s and [[reduced ring]]s.

Most definitions and assertions in this article appear in {{harv|Lam|1999}} and {{harv|Lam|2001}}.

==Definitions==

For a commutative ring ''R'', a proper ideal ''A'' is a '''semiprime ideal''' if ''A'' satisfies either of the following equivalent conditions:
* If ''x''''k'' is in ''A'' for some positive integer ''k'' and element ''x'' of ''R'', then ''x'' is in ''A''.
* If ''y'' is in ''R'' but not in ''A'', all positive integer powers of ''y'' are not in ''A''.

The latter condition that the complement is "closed under powers" is analogous to the fact that complements of prime ideals are closed under multiplication.

As with prime ideals, this is extended to noncommutative rings "ideal-wise". The following conditions are equivalent definitions for a semiprime ideal ''A'' in a ring ''R'':
* For any ideal ''J'' of ''R'', if ''J''''k''⊆''A'' for a positive natural number ''k'', then ''J''⊆''A''.
* For any ''right'' ideal ''J'' of ''R'', if ''J''''k''⊆''A'' for a positive natural number ''k'', then ''J''⊆''A''.
* For any ''left'' ideal ''J'' of ''R'', if ''J''''k''⊆''A'' for a positive natural number ''k'', then ''J''⊆''A''.
* For any ''x'' in ''R'', if ''xRx''⊆''A'', then ''x'' is in ''A''.

Here again, there is a noncommutative analogue of prime ideals as complements of [[Prime_ideal#Prime_ideals_for_noncommutative_rings|m-systems]]. A nonempty subset ''S'' of a ring ''R'' is called an '''n-system''' if for any ''s'' in ''S'', there exists an ''r'' in ''R'' such that ''srs'' is in ''S''. With this notion, an additional equivalent point may be added to the above list:

* ''R''\''A'' is an n-system.

The ring ''R'' is called a '''semiprime ring''' if the zero ideal is a semiprime ideal. In the commutative case, this is equivalent to ''R'' being a [[reduced ring]], since ''R'' has no nonzero nilpotent elements. In the noncommutative case, the ring merely has no nonzero nilpotent right ideals. So while a reduced ring is always semiprime, the converse is not true.<ref>The full ring of two-by-two matrices over a field is semiprime with nonzero nilpotent elements.</ref>

==General properties of semiprime ideals==

To begin with, it is clear that prime ideals are semiprime, and that for commutative rings, a semiprime [[primary ideal]] is prime.

While the intersection of prime ideals is not usually prime, it ''is'' a semiprime ideal. Shortly it will be shown that the converse is also true, that every semiprime ideal is the intersection of a family of prime ideals.

For any ideal ''B'' in a ring ''R'', we can form the following sets:
:<math>\sqrt{B}:=\bigcap\{ P\subseteq R \mid B \subseteq P, P \mbox{ a prime ideal} \}\subseteq\{x\in R\mid x^n\in B \mbox{ for some }k\in\mathbb{N}^+ \} \,</math>

The set <math>\sqrt{B}</math> is the definition of the [[radical of an ideal|radical of ''B'']] and is clearly a semiprime ideal containing ''B'', and in fact is the smallest semiprime ideal containing ''B''. The inclusion above is sometimes proper in the general case, but for commutative rings it becomes an equality.

With this definition, an ideal ''A'' is semiprime if and only if <math>\sqrt{A}=A</math>. At this point, it is also apparent that every semiprime ideal is in fact the intersection of a family of prime ideals. Moreover, this shows that the intersection of any two semiprime ideals is again semiprime.

By definition ''R'' is semiprime if and only if <math>\sqrt{\{0\}}=\{0\}</math>, that is, the intersection of all prime ideals is zero. This ideal <math>\sqrt{\{0\}}</math> is also denoted by <math>Nil_*(R)\,</math> and also called '''Baer's lower [[nilradical of a ring|nilradical]]''' or the '''Baer-Mccoy radical''' or the '''prime radical''' of ''R''.

==Semiprime Goldie rings==

{{Empty section|date=July 2012}}
{{main|Goldie ring}}

==References==
{{Reflist}}
*{{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | mr=1653294 | year=1999}}

*{{citation |author=Lam, T. Y. |title=A first course in noncommutative rings |series=Graduate Texts in Mathematics |volume=131 |edition=2 |publisher=Springer-Verlag |place=New York |year=2001 |pages=xx+385 |isbn=0-387-95183-0 |mr=1838439 }}

==External links==
* [http://planetmath.org/encyclopedia/SemiprimeIdeal.html PlanetMath article on semiprime ideals]

[[Category:Ring theory]]
[[Category:Ideals]]

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