Serial module

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 N1 and N2 of M, either N 1 ⊆ N 2 {displaystyle N_{1}subseteq N_{2}} or N 2 ⊆ N 1 {displaystyle N_{2}subseteq N_{1}} . 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. 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 N1 and N2 of M, either N 1 ⊆ N 2 {displaystyle N_{1}subseteq N_{2}} or N 2 ⊆ N 1 {displaystyle N_{2}subseteq N_{1}} . 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 Z / n Z {displaystyle mathbb {Z} /nmathbb {Z} } for any integer n > 1 {displaystyle n>1} . 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 (Puninski 2001) and (Hazewinkel 2004). 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. 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 R = ⊕ i = 1 n e i R {displaystyle R=oplus _{i=1}^{n}e_{i}R} where each ei is an idempotent element and eiR 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