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In abstract algebra, a uniserial module ''M'' is a module over a ring ''R'', whose submodules are totally ordered by inclusion. This means simply that for any two submodules ''N''1 and ''N''2 of ''M'', either or . A module is called a serial module if it is a direct sum of uniserial modules. A ring ''R'' is called a right uniserial ring if it is uniserial as a right module over itself, and likewise called a right serial ring if it is a right serial module over itself. Left uniserial and left serial rings are defined in an analogous way, and are in general distinct from their right counterparts. An easy motivational example is the quotient ring for any integer . This ring is always serial, and is uniserial when ''n'' is a prime power. The term ''uniserial'' has been used differently from the above definition: for clarification see this section. A partial alphabetical list of important contributors to the theory of serial rings includes the mathematicians Keizo Asano, I. S. Cohen, P.M. Cohn, Yu. Drozd, D. Eisenbud, A. Facchini, A.W. Goldie, Phillip Griffith, I. Kaplansky, V.V Kirichenko, G. Köthe, H. Kuppisch, I. Murase, T. Nakayama, P. Příhoda, G. Puninski, and R. Warfield. References for each author can be found in and . Following the common ring theoretic convention, if a left/right dependent condition is given without mention of a side (for example, uniserial, serial, Artinian, Noetherian) then it is assumed the condition holds on both the left and right. Unless otherwise specified, each ring in this article is a ring with unity, and each module is unital. ==Properties of uniserial and serial rings and modules== It is immediate that in a uniserial ''R''-module ''M'', all submodules except ''M'' and 0 are simultaneously essential and superfluous. If ''M'' has a maximal submodule, then ''M'' is a local module. ''M'' is also clearly a uniform module and thus is directly indecomposable. It is also easy to see that every finitely generated submodule of ''M'' can be generated by a single element, and so ''M'' is a Bézout module. It is known that the endomorphism ring EndR(''M'') is a semilocal ring which is very close to a local ring in the sense that EndR(''M'') has at most two maximal right ideals. If ''M'' is required to be Artinian or Noetherian, then EndR(''M'') is a local ring. Since rings with unity always have a maximal right ideal, a right uniserial ring is necessarily local. As noted before, a finitely generated right ideal can be generated by a single element, and so right uniserial rings are right Bézout rings. A right serial ring ''R'' necessarily factors in the form where each ''e''i is an idempotent element and ''e''i''R'' is a local, uniserial module. This indicates that ''R'' is also a semiperfect ring, which is a stronger condition than being a semilocal ring. Köthe showed that the modules of Artinian principal ideal rings (which are a special case of serial rings) are direct sums of cyclic submodules. Later, Cohen and Kaplansky determined that a commutative ring ''R'' has this property for its modules if and only if ''R'' is an Artinian principal ideal ring. Nakayama showed that Artinian serial rings have this property on their modules, and that the converse is not true The most general result, perhaps, on the modules of a serial ring is attributed to Drozd and Warfield: it states that every finitely presented module over a serial ring is a direct sum of cyclic uniserial submodules (and hence is serial). If additionally the ring is assumed to be Noetherian, the finitely presented and finitely generated modules coincide, and so all finitely generated modules are serial. Being right serial is preserved under direct products of rings and modules, and preserved under quotients of rings. Being uniserial is preserved for quotients of rings and modules, but never for products. A direct summand of a serial module is not necessarily serial, as was proved by Puninski, but direct summands of ''finite'' direct sums of uniserial modules are serial modules . It has been verified that Jacobson's conjecture holds in Noetherian serial rings. 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「serial module」の詳細全文を読む スポンサード リンク
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