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• W56892391 abstract "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." @default.
• W56892391 created "2016-06-24" @default.
• W56892391 creator A5057554809 @default.
• W56892391 date "2002-01-01" @default.
• W56892391 modified "2023-09-25" @default.
• W56892391 cites W124419101 @default.
• W56892391 cites W1490790829 @default.
• W56892391 cites W1496544353 @default.
• W56892391 cites W1497483297 @default.
• W56892391 cites W1504365199 @default.
• W56892391 cites W1512190817 @default.
• W56892391 cites W1537584867 @default.
• W56892391 cites W1558352842 @default.
• W56892391 cites W1581708723 @default.
• W56892391 cites W1590547289 @default.
• W56892391 cites W1591439603 @default.
• W56892391 cites W1603686988 @default.
• W56892391 cites W181219113 @default.
• W56892391 cites W1917961700 @default.
• W56892391 cites W1934267642 @default.
• W56892391 cites W1963547508 @default.
• W56892391 cites W1964626742 @default.
• W56892391 cites W1965014670 @default.
• W56892391 cites W1969530627 @default.
• W56892391 cites W1974168323 @default.
• W56892391 cites W1982089796 @default.
• W56892391 cites W1984362391 @default.
• W56892391 cites W1985612074 @default.
• W56892391 cites W1988579890 @default.
• W56892391 cites W1989346996 @default.
• W56892391 cites W1995579478 @default.
• W56892391 cites W1999110370 @default.
• W56892391 cites W2001959590 @default.
• W56892391 cites W2005840455 @default.
• W56892391 cites W2010334894 @default.
• W56892391 cites W2011237288 @default.
• W56892391 cites W2011331350 @default.
• W56892391 cites W2013182085 @default.
• W56892391 cites W2014599029 @default.
• W56892391 cites W2015167102 @default.
• W56892391 cites W2016522391 @default.
• W56892391 cites W2022584270 @default.
• W56892391 cites W2028555493 @default.
• W56892391 cites W2037735261 @default.
• W56892391 cites W2038414314 @default.
• W56892391 cites W2044041144 @default.
• W56892391 cites W2046327176 @default.
• W56892391 cites W2050154447 @default.
• W56892391 cites W2051261473 @default.
• W56892391 cites W2053709301 @default.
• W56892391 cites W2053905484 @default.
• W56892391 cites W2057345209 @default.
• W56892391 cites W2059705546 @default.
• W56892391 cites W2060660845 @default.
• W56892391 cites W2061134683 @default.
• W56892391 cites W2067621580 @default.
• W56892391 cites W2070944369 @default.
• W56892391 cites W2071476325 @default.
• W56892391 cites W2080813428 @default.
• W56892391 cites W2084343509 @default.
• W56892391 cites W2084715556 @default.
• W56892391 cites W2085279618 @default.
• W56892391 cites W2085860546 @default.
• W56892391 cites W2086131079 @default.
• W56892391 cites W2090927985 @default.
• W56892391 cites W2091983059 @default.
• W56892391 cites W2093814461 @default.
• W56892391 cites W2096382743 @default.
• W56892391 cites W2099916937 @default.
• W56892391 cites W2117886859 @default.
• W56892391 cites W2124599666 @default.
• W56892391 cites W2126036357 @default.
• W56892391 cites W2140235066 @default.
• W56892391 cites W2144236580 @default.
• W56892391 cites W2148582132 @default.
• W56892391 cites W2162397735 @default.
• W56892391 cites W2320385936 @default.
• W56892391 cites W2322922765 @default.
• W56892391 cites W2323323566 @default.
• W56892391 cites W2596022583 @default.
• W56892391 cites W2724651846 @default.
• W56892391 cites W2735169764 @default.
• W56892391 cites W2737987007 @default.
• W56892391 cites W569990221 @default.
• W56892391 cites W629470101 @default.
• W56892391 cites W87187299 @default.
• W56892391 cites W985604590 @default.
• W56892391 cites W1573118306 @default.
• W56892391 doi "https://doi.org/10.1023/a:1014958025082" @default.
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