|
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many ''primary ideals'' (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by . The Lasker–Noether theorem is an extension of the fundamental theorem of arithmetic, and more generally the fundamental theorem of finitely generated abelian groups to all Noetherian rings. The Lasker–Noether theorem plays an important role in algebraic geometry, by asserting that every algebraic set may be uniquely decomposed into a finite union of irreducible components. It has a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules. This contains the case for rings as a special case, considering the ring as a module over itself, so that ideals are submodules. This also generalizes the primary decomposition form of the structure theorem for finitely generated modules over a principal ideal domain, and for the special case of polynomial rings over a field, it generalizes the decomposition of an algebraic set into a finite union of (irreducible) varieties. The first algorithm for computing primary decompositions for polynomial rings was published by Noether's student . ==Definitions== Write ''R'' for a commutative ring, and ''M'' and ''N'' for modules over it. * A zero divisor of a module ''M'' is an element ''x'' of ''R'' such that ''xm'' = 0 for some non-zero ''m'' in ''M''. *An element ''x'' of ''R'' is called nilpotent in ''M'' if ''x''''n''''M'' = 0 for some positive integer ''n''. *A module is called coprimary if every zero divisor of ''M'' is nilpotent in ''M''. For example, groups of prime power order and free abelian groups are coprimary modules over the ring of integers. *A submodule ''M'' of a module ''N'' is called a primary submodule if ''N''/''M'' is coprimary. *An ideal ''I'' is called primary if it is a primary submodule of ''R''. This is equivalent to saying that if ''ab'' is in ''I'' then either ''a'' is in ''I'' or ''b''''n'' is in ''I'' for some ''n'', and to the condition that every zero-divisor of the ring ''R''/''I'' is nilpotent. *A submodule ''M'' of a module ''N'' is called irreducible if it is not an intersection of two strictly larger submodules. *An associated prime of a module ''M'' is a prime ideal that is the annihilator of some element of ''M''. 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「Primary decomposition」の詳細全文を読む スポンサード リンク
|