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 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 motivating 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 below. 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. 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.