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In abstract algebra, an associated prime of a module ''M'' over a ring ''R'' is a type of prime ideal of ''R'' that arises as an annihilator of a (prime) submodule of ''M''. The set of associated primes is usually denoted by . In commutative algebra, associated primes are linked to the Lasker-Noether primary decomposition of ideals in commutative Noetherian rings. Specifically, if an ideal ''J'' is decomposed as a finite intersection of primary ideals, the radicals of these primary ideals are prime ideals, and this set of prime ideals coincides with . Also linked with the concept of "associated primes" of the ideal are the notions of isolated primes and embedded primes. ==Definitions== A nonzero ''R'' module ''N'' is called a prime module if the annihilator for any nonzero submodule ''N' '' of ''N''. For a prime module ''N'', is a prime ideal in ''R''. An associated prime of an ''R'' module ''M'' is an ideal of the form where ''N'' is a prime submodule of ''M''. In commutative algebra the usual definition is different, but equivalent: if ''R'' is commutative, an associated prime ''P'' of ''M'' is a prime ideal of the form for a nonzero element ''m'' of ''M'' or equivalently is isomorphic to a submodule of ''M''. In a commutative ring ''R'', minimal elements in (with respect to the set-theoretic inclusion) are called isolated primes while the rest of the associated primes (i.e., those properly containing associated primes) are called embedded primes. A module is called coprimary if ''xm'' = 0 for some nonzero ''m'' ∈ ''M'' implies ''x''''n''''M'' = 0 for some positive integer ''n''. A nonzero finitely generated module ''M'' over a commutative Noetherian ring is coprimary if and only if it has exactly one associated prime. A submodule ''N'' of ''M'' is called ''P''-primary if is coprimary with ''P''. An ideal ''I'' is a ''P''-primary ideal if and only if ; thus, the notion is a generalization of a primary ideal. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「associated prime」の詳細全文を読む スポンサード リンク
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