All rings are assumed to be associative and (except for nil-rings and for some stipulated cases) to have nonzero identity elements. A ring A is an exchange ring if the following two equivalent conditions hold: (1) for any element a ∈ A, there exists an idempotent e ∈ aA with 1 − e ∈ (1 − a)A; (2) for any element a ∈ A, there exists an idempotent e ∈ Aa with 1− e ∈ A(1− a). (The equivalence (1)⇐⇒(2) is proved in 2.1 below.) A ring A is said to be regular if for every element a of A, there exists an element b of A such that a = aba. A ring A is semiregular if A/J(A) is a regular ring and all idempotents of A/J(A) can be lifted to idempotents of A. A ring A is a π-regular ring if, for every element a of A, there exists an element b of A such that a = aba for some positive integer n. Every semiregular or π-regular ring is an exchange ring (see 2.11(1)). It is directly verified that the ring of all rational numbers with odd denominators is an exchange ring that is not a π-regular ring. Let B be the ring of all rational numbers with odd denominators, {Ai}i=1 be a countable infinite set of copies of the field of rational numbers, D be the direct product of all rings Ai, and R be the subring in D generated by the ideal ⊕i=1Ai and by the subring {(b, b, b, . . . ) | b ∈ B}. Then R is a commutative reduced semiprimitive exchange ring that is not a semiregular ring (see 2.9(4)). A moduleM has the finite exchange property if for every moduleX and for every direct decomposition X = M ′ ⊕ Y = ⊕i∈INi, where M ′ ∼= M and I is a finite set, there exist submodules N ′ i ⊆ Ni (i ∈ I) such that X = M ′ ⊕ (⊕i∈IN ′ i). The importance of exchange rings is related to the fact that a module M has the finite exchange property if and only if the ring End(M) is an exchange ring [75]. The systematic study of modules with the finite exchange property and exchange rings was initiated in [18,58,75]. Also, see [2–4,14,17,39,43,48,55,56,58–61,63,67,73–75,77–79,82–85,87–90,93]. Exchange rings are potent rings (a ring A is potent if every right ideal of A that is not contained in J(A) contains a nonzero idempotent and all idempotents of A/J(A) can be lifted to idempotents of A). Thus, exchange rings have many idempotents. For a module M , the Jacobson radical, the endomorphism ring, and the lattice of all submodules are denoted by J(M), End(M), and Lat(M), respectively. For a ring A, C(A) and U(A) denote the center and the group of invertible elements of A, respectively. For a subset B of a ring A, the right annihilator and the left annihilator of B in A are denoted by rA(B) and A(B), respectively. We can omit the subscripts if the situation is obvious. If X and Y are subsets of a ring A, then XY denotes the set of all finite sums { ∑ xiyi | xi ∈ X, yi ∈ Y }. In particular, ABA denotes the ideal of A generated by its subset B. A submoduleN of a moduleM is essential (in M) if N has the nonzero intersection with any nonzero submodule of M . In this case, we say that the module M is an essential extension of N . A right (resp. left) module M over a ring A is a nonsingular module if r(m) (resp. (m)) is not an essential right (resp. left) ideal of A for every nonzero element m of M . A ring without nonzero nilpotent elements is called a reduced ring. A ring is normal if all its idempotents are central. Every reduced ring A is normal, since for any idempotent e ∈ A, we have 0 = (eA(1 − e))2 = ((1− e)Ae)2, whence eA(1− e) = (1− e)Ae = 0.
Read full abstract
Dec 1, 2018
Open Access