ABSTRACT ALGEBRA ON LINE: Structure of Noncommutative Rings

STRUCTURE OF NONCOMMUTATIVE RINGS

11.1 Prime and primitive ideals

11.2 The Jacobson radical

11.3 Semisimple Artinian rings

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Prime and primitive ideals

11.1.1. Definition. A proper ideal P of the ring R is called a prime ideal if AB is contained in

P implies A is contained in

P or B

P, for any ideals A, B of R.
A proper ideal I of the ring R is called a semiprime ideal if it is an intersection of prime ideals of R.
A proper ideal P of the ring R is called a left primitive ideal if it is the annihilator of a simple left R-module.

11.1.2. Definition. The ring R is called a simple ring if (0) is a maximal ideal; it is called a prime ring if (0) is a prime ideal, and a semiprime ring if (0) is a semiprime ideal. Finally, R is said to be a (left) primitive ring if (0) is a primitive ideal.

11.1.3. Proposition. The following conditions are equivalent for the proper ideal P of the ring R:
(1) P is a prime ideal;
(2) AB is contained in P implies A P or B P, for any ideals A, B of R which contain P;
(3) AB P implies A = P or B = P, for any left ideals A, B of R with P A and P B;
(4) aRb P implies a is in P or b is in P, for any a,b R.

11.1.4. Definition. The nonzero module _RM is called a prime module if AN = (0) implies N = (0) or AM = (0), for any ideal A of R and any submodule N of M.

11.1.5. Proposition. The annihilator of a prime module is a prime ideal.

11.1.6. Proposition. Any maximal ideal is primitive, and any primitive ideal is prime.
In a left Artinian ring, the notions of maximal ideal, primitive ideal, and prime ideal coincide.

An ideal I of the ring R is said to be nilpotent if Aⁿ = (0) for some positive integer n. It is said to be a nil ideal if for each a I there exists a positive integer n such that aⁿ = 0.

11.1.7. Proposition. The following conditions are equivalent for the proper ideal I of the ring R:
(1) I is a semiprime ideal;
(2) the ring R/I has no nonzero nilpotent ideals;
(3) AB is contained in I implies A intersect B I, for any ideals A, B of R;
(4) AB I implies AB = I, for any left ideals A, B of R with I A and I B;
(5) aRa I implies a is in I, for all a R.

11.1.8. Definition. Let _RM be a left R-module. The bicommutator of M is the subring of End_Z(M) defined by

Bic_R(M) = { alpha End_Z(M) | f alpha = alpha f for all f End_R(M) }.

11.1.9. Lemma. If M is any left R-module, then R/Ann(M) is isomorphic to a subring of Bic_R(M).

11.1.10. Proposition. If P is a primitive ideal of the ring R, then there exists a division ring D and a vector space V over D for which R/P is isomorphic to a subring of the ring of all linear transformations from V into V.

11.1.11. Proposition. Let M be a left R-module.
(a) The endomorphism ring End_R(Mⁿ) is isomorphic to the ring of n×n matrices with entries in End_R(M).
(b) The bicommutator Bic_R(Mⁿ) is isomorphic to Bic_R(M).

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 N intersect K is not equal to (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.

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 rho (M) such that
(i) f( rho (M)) is a subset of rho (N), for all modules _RN and all f Hom_R(M,N);
(ii) rho (M/ rho (M)) = (0).

11.2.6. Definition. Let C be any class of left R-modules. For any module _RM we make the following definition.
rad_C(M) = the intersection of ker(f),
where the intersection is taken over all R-homomorphisms f : M -> X, for all X in C.

11.2.7. Proposition. Let rho be a radical for the class of left R-modules, and let F be the class of left R-modules X for which rho (X) = (0).
(a) rho (R) is a two-sided ideal of R.
(b) rho (R) M is a subset of rho (M) for all modules _RM.
(c) rad_F is a radical, and rho = rad_F.
(d) rho (R) = the intersection of 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.

Semisimple Artinian rings

11.3.1. Theorem. Any simple ring with a minimal left ideal is isomorphic to a ring of n×n matrices over a division ring.

11.3.2. Theorem. [Artin-Wedderburn] The following conditions are equivalent for a ring R with identity.
(1) R is left Artinian and J(R) = (0);
(2) _RR is a semisimple module;
(3) R is isomorphic to a finite direct product of rings of n×n matrices over division rings.

11.3.3. Definition. A ring which satisfies the conditions of the previous theorem is said to be semisimple Artinian.

11.3.4. Corollary. The following conditions are equivalent for a ring R with identity.
(1) R is semisimple Artinian;
(2) R is left Artinian and semiprime;
(3) every left R-module is completely reducible;
(4) every left R-module is projective;
(5) every left R-module is injective.

11.3.5. Theorem. [Hopkins] Any left Artinian ring is left Noetherian.

11.3.6. Lemma. Let _RM be a simple module, and let D = End_R(M). If _DV is any finite dimensional subspace of _DM, then

V = { m M | Ann_R(V) m = 0 }.

Let V be a left vector space over a division ring D. A subring R is called a dense subring of End_D(V) if for each n>0, each linearly independent subset {u₁,u₂,...,u_n} of V, and each arbitrary subset {v₁,v₂,...,v_n} of V, there exists an element theta

R such that theta

(u_i) = v_i for all i = 1,...,n.

11.3.7. Theorem. [Jacobson Density Theorem] Any (left) primitive ring is isomorphic to a dense ring of linear transformations of a vector space over a division ring.

11.3.8. Proposition. If _RM is completely reducible and finitely generated, then End_R(M) is isomorphic to a finite direct product of rings of n×n matrices over division rings.

11.3.9. Definition. A ring R is called von Neumann regular if for each a R there exists b R such that aba = a.

11.3.10. Proposition. If _RM is completely reducible, then End_R(M) is von Neumann regular.

11.3.11. Proposition. The following conditions are equivalent for a ring R.
(1) R is von Neumann regular;
(2) each principal left ideal of R is generated by an idempotent element;
(3) each finitely generated left ideal of R is generated by an idempotent element.

11.1.12. Proposition. If R is a von Neumann regular ring, then J(R) = (0).

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