Simple Module
In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory.
In this article, all modules will be assumed to be right unital modules over a ring R.
Read more about Simple Module: Examples, Basic Properties of Simple Modules, Simple Modules and Composition Series, The Jacobson Density Theorem
Famous quotes containing the word simple:
“Meantime the education of the general mind never stops. The reveries of the true and simple are prophetic. What the tender poetic youth dreams, and prays, and paints today, but shuns the ridicule of saying aloud, shall presently be the resolutions of public bodies, then shall be carried as grievance and bill of rights through conflict and war, and then shall be triumphant law and establishment for a hundred years, until it gives place, in turn, to new prayers and pictures.”
—Ralph Waldo Emerson (1803–1882)