The Extended Centralizer of a Ring over a Module
نویسندگان
چکیده
In a recent paper,1 K. Asano gave a new proof of the theorem that a domain of integrity has a right quotient ring if and only if every pair of nonzero elements has a common nonzero right multiple. His method of proof is used in the present work to extend the centralizer of a ring over a module to a system of semi-endomorphisms of the module. From this extension, necessary and sufficient conditions that a ring have a right quotient regular ring are derived. Consider a given ring R, and a given nonzero right i?-moduIe M. Denote by 3JÎ the set of all submodules of M, and by SDÎ* the set of all submodules N oí M having the property that NÍ^N't^O for all nonzero TV'GSOîSince ME^R*, SDÎ* is not void. It is easily seen that if N and N' are in M*, then N+N' and NC\N' are also in Stt*. Thus {9ÏÏ*; Q, r\, +} is a sublattice of the lattice {W; Q, C\, + }. An i?-homomorphism of N into M, N any element of SO?, is called a semi-endomorphism of M. Thus, thinking of the semi-endomorphism a as a left operator on N, we have a(x+y) =ax+ay and a(xa) = (ax)a for all x, yEN, aERFor convenience, the module N on which a is defined is denoted by Ma. The set of all semi-endomorphisms of M is labeled with 31. Contained in 21 is the usual centralizer of R over M consisting of all a E 21 for which Ma = M. A partial ordering = is defined in 2Í as follows: a^ß if and only if MaÇZMp and ax=ßx for all xEMa. The notation a<ß is used in case a=ß and Maj^Mß. In case ? is a linearly ordered subset of 21, and M' = [)Ma, «GS. the mapping 7 of M' into M defined by
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