Artinian Module - Left and Right Artinian Rings, Modules and Bimodules

Left and Right Artinian Rings, Modules and Bimodules

The ring R can be considered as a right module, where the action is the natural one given by the ring multiplication on the right. R is called right Artinian when this right module R is an Artinian module. The definition of "left Artinian ring" is done analogously. For noncommutative rings this distinction is necessary, because it is possible for a ring to be Artinian on one side only.

The left-right adjectives are not normally necessary for modules, because the module M is usually given as a left or right R module at the outset. However, it is possible that M may have both a left and right R module structure, and then calling M Artinian is ambiguous, and it becomes necessary to clarify which module structure is Artinian. To separate the properties of the two structures, one can abuse terminology and refer to M as left Artinian or right Artinian when, strictly speaking, it is correct to say that M, with its left R-module structure, is Artinian.

The occurrence of modules with a left and right structure is not unusual: for example R itself has a left and right R module structure. In fact this is an example of a bimodule, and it may be possible for an abelian group M to be made into a left-R, right-S bimodule for a different ring S. Indeed, for any right module M, it is automatically a left module over the ring of integers Z, and moreover is a Z-R bimodule. For example, consider the rational numbers Q as a Z-Q bimodule in the natural way. Then Q is not Artinian as a left Z module, but it is Artinian a right Q module.

The Artinian condition can be defined on bimodule structures as well: an Artinian bimodule is a bimodule whose poset of sub-bimodules satisfies the descending chain condition. Since a sub-bimodule of an R-S bimodule M is a fortiori a left R-module, if M considered as a left R module were Artinian, then M is automatically an Artinian bimodule. It may happen, however, that a bimodule is Artinian without its left or right structures being Artinian, as the following example will show.

Example: It is well known that a simple ring is left Artinian if and only if it is right Artinian, in which case it is a semisimple ring. Let R be a simple ring which is not right Artinian. Then it is also not left Artinian. Considering R as an R-R bimodule in the natural way, its sub-bimodules are exactly the ideals of R. Since R is simple there are only two: R and the zero ideal. Thus the bimodule R is Artinian as a bimodule, but not Artinian as a left or right R-module over itself.