The Jacobson radical
11.2.1.Definition.Let M be a left R-module.The intersection of all maximal submodules of M is called the Jacobson radical of M,and is denoted by J(M).
11.2.2.Definition.Let M be a left R-module.
The submodule N of M is called essential or large in M if NK (0) for all nonzero submodules K of M.
The submodule N is called superfluous or small in M if N+K M for all proper submodules K of M.
Kernel
Definition Let :R->S be a ring homomorphism.The set
{ a R | (a) = 0 }
is called the kernel of ,denoted by ker().
11.2.3.Proposition.Let N be a submodule of RM.If K is maximal in the set of all submodules of M that have trivial intersection with N,then N+K is essential in M,and (N+K)/K is essential in M/K.
11.2.4.Proposition.The socle of any module is the intersection of its essential submodules.
11.2.5.Definition.A radical for the class of left R-modules is a function that assigns to each module RM a submodule (M) such that
(i) f((M)) (N),for all modules RN and all f HomR(M,N);
(ii) (M/(M)) = (0).
11.2.6.Definition.Let C be any class of left R-modules.For any module RM we make the following definition.
radC(M) = ker(f),
where the intersection is taken over all R-homomorphisms f :M -> X,for all X in C.
11.2.7.Proposition.Let be a radical for the class of left R-modules,and let F be the class of left R-modules X for which (X) = (0).
(a) (R) is a two-sided ideal of R.
(b) (R) M (M) for all modules RM.
(c) radF is a radical,and = radF.
(d) (R) = Ann(X),where the intersection is taken over all modules X in F.
11.2.8.Lemma.[Nakayama] If R M is finitely generated and J(R)M = M,then M = (0).
11.2.9.Proposition.Let M be a left R-module.
(a) J(M) = { m M | Rm is small in M }.
(b) J(M) is the sum of all small submodules of M.
(c) If M is finitely generated,then J(M) is a small submodule.
(d) If M is finitely generated,then M/J(M) is semisimple if and only if it is Artinian.
11.2.10.Theorem.The Jacobson radical J(R) of the ring R is equal to each of the following sets:
(1) The intersection of all maximal left ideals of R;
(2) The intersection of all maximal right ideals of R;
(3) The intersection of all left-primitive ideals of R;
(4) The intersection of all right-primitive ideals of R;
(5) { x R | 1-ax is left invertible for all a R };
(6) { x R | 1-xa is right invertible for all a R };
(7) The largest ideal J of R such that 1-x is invertible in R for all x J.
11.2.11.Definition.The ring R is said to be semiprimitive if J(R) = (0).
11.2.12.Proposition.Let R be any ring.
(a) The Jacobson radical of R contains every nil ideal of R.
(b) If R is left Artinian,then the Jacobson radical of R is nilpotent.