Comultiplication Modules over Noncommutative Rings

Abstract

Comultiplication modules over not necessarily commutative rings are studied. All rings are assumed to be associative and with nonzero identity element; all modules are assumed to be unitary. Expressions of the form “A is an invariant ring” mean that “AA and AA are right and left invariant rings.” If A is a ring and M is a right A-module, then for any subset B in A, we denote by lM (B) the left annihilator {m ∈ M | mB = 0} of B in M . According to [2], a module MA is called a comultiplication module if for any submodule X of M , there exists an ideal B of the ring A such that X = lM (B). Comultiplication modules are studied in many papers (see, e.g., [1–6]), where the focus is mainly on comultiplication modules over commutative rings. Theorem 1 ([1]). Let A be a commutative ring and let M be a comultiplication A-module. (1) M is an essential extension of a direct sum of pairwise nonisomorphic simple modules, and any submodule of the module M that is a finite direct sum of cyclic modules is a cyclic module. (2) If the module M is nonsingular, then M is a projective semisimple module. In connection with Theorem 1, we prove Theorem 2, which is the main result of the present paper. Theorem 2. Let A be a ring and let M be a comultiplication right A-module. (1) Any submodule of the module M that is a finite direct sum of cyclic modules is a cyclic module. (2) If A is an invariant ring with commutative multiplication of ideals, then M is an essential extension of a direct sum of pairwise nonisomorphic simple modules; in addition, if the module M is nonsingular, then M is a projective semisimple module. In connection with Theorem 2, we note that the class of all invariant rings with commutative multiplication of ideals contains all commutative rings, all rings of formal power series in one variable over division rings, all factor rings of any direct products of division rings, and all strongly regular rings. (A ring A is said to be strongly regular if every principal one-sided ideal of A is generated by a central idempotent.) The proof of Theorem 2 is decomposed into a series of assertions; some of the assertions are of independent interest. We present the necessary notation and definitions. A ring A is said to be right invariant if all right ideals of A are ideals. A ring A is said to be dual if B = rA(lA)(B) for any right ideal B of A and C = lA(rA)(C) for any left ideal C of A. We denote by J(M) the Jacobson radical of the module M , i.e., J(M) is the intersection of all maximal submodules in M ; we have J(M) = M if M does not have maximal submodules. We denote by Soc(M) the socle of the module M , i.e., Soc(M) is the sum of all minimal submodules in M , and Soc(M) = 0 if M does not have minimal submodules. A ring A is said to be semiperfect if A/J(A) is an Artinian ring and all idempotents of A/J(A) are lifted to idempotents of the ring A. A cyclic module M is said to be local if the module M/J(M) is simple. A submodule X of the module M is said to be essential if X has the nonzero intersection with every nonzero submodule in M . In this case, we say that M is an essential extension of the module X. A module MA is said to be nonsingular if M does not have nonzero elements whose annihilators are essential right ideals of the ring A. A module M is said to be cocyclic if M is an essential extension of a simple module. A module M is said to be finite-dimensional if M does not contain an infinite direct sum of nonzero submodules. A module M is said to be quotient finite-dimensional if all factor modules Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 4, pp. 217–224, 2011/12. 1072–3374/13/1915–0743 c © 2013 Springer Science+Business Media New York 743 of the module M are finite-dimensional. For a module M , a submodule X in M is said to be superfluous (in M) if X + Y = M for each proper submodule Y in M . Any submodule of a factor module of M is called a subfactor of M . A module MA is said to be faithful if rA(M) = 0. The following two assertions are known; they can be directly verified. Lemma 3 ([1, Lemma 1.2]). Let A be a ring, B be an ideal in A, and let M be a right A-module with MB = 0. Then M is a comultiplication A-module if and only if M is a comultiplication A/B-module. Lemma 4 ([2, Theorem 3.17, Lemma 3.7]). Let A be a ring and let M be a right A-module. The following conditions are equivalent : (1) M is a comultiplication module; (2) X = lM ( rA(X) ) for every submodule X of the module M ; (3) every submodule of the module M is a comultiplication module. Lemma 5. Let A be a ring, M be a comultiplication right A-module, and let X be a nonzero submodule in M . (1) X is a comultiplication A/rA(X)-module and f(X) ⊆ X for any homomorphism f : X → M . In particular, M does not contain a direct sum of two isomorphic nonzero modules. (2) If X = X1⊕· · ·⊕Xn is a submodule in M and there exists a module N such that every module Xi is a homomorphic image of the module N , then X is a homomorphic image of the module N . (3) If X = X1 ⊕ · · · ⊕ Xn is a submodule in M and every module Xi is cyclic, then X is a cyclic module. (4) In M , every finitely generated semisimple submodule is cyclic. (5) If there exists an element m ∈ M with rA(m) = 0, then A is a comultiplication right A-module. (6) If the ring A is right invariant and M contains a faithful cyclic submodule mA, then A is a comultiplication right A-module. (7) If the module M is nonsingular and the multiplication of right ideals of the ring A is commutative, then M is a projective semisimple module. Proof. (1) By Lemma 4(3) and Lemma 3, X is a comultiplication A/rA(X)-module. Since M is a comultiplication module, X = rM (B) for some ideal B in A. Then f(X)B ⊆ f(XB) = f(0) = 0, whence f(X) ⊆ rM (B) = X. (2) By assumption, there exist epimorphisms hi : N → Xi, i = 1, . . . , n. We denote by h the homomorphism h1 + · · · + hn from N into X = X1 ⊕ · · · ⊕Xn. Let πi be the natural projection from the module X onto the module Xi, i = 1, . . . , n. By (1), πi ( h(N) ) ⊆ h(N) for all i. Therefore, h(N) = π1 ( h(N) ⊕ · · · ⊕ πn ( h(N) ) = X1 ⊕ · · · ⊕Xn = X, and h is the required epimorphism from the module N onto the module X. (3) Since every module Xi is cyclic, there exist epimorphisms hi : AA → Xi, i = 1, . . . , n. By (2), there exists an epimorphism from the module AA onto the module X. Therefore, X is a cyclic module. (4) Since every finitely generated semisimple module is a finite direct sum of simple modules, the assertion follows from (3) and the property that every simple module is cyclic. (5) By Lemma 4, mAA is a comultiplication module. In addition, mAA is a free cyclic module with free generator m. Now it is directly verified that AA is a comultiplication module. (6) Since the ring A is right invariant, rA(mA) = rA(m). Therefore, (6) follows from (5). (7) First, we prove that M is a semisimple module. Let Y be a submodule in M . By Zorn’s lemma, there exists a submodule Z in M such that Y ∩Z = 0 and Y ⊕Z is an essential submodule in M . We set N = Y ⊕ Z. It is sufficient to prove that M = N . We assume the contrary. Then there exists a nonzero element m ∈ M \ N . Since M is a comultiplication module, there exists an ideal B of the ring A with N = lM (B). Then mB = 0. Since N is an essential submodule in M , we have that mC ⊆ N for some