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Inverses and zero-divisors

1942; American Mathematical Society; Volume: 48; Issue: 8 Linguagem: Inglês

10.1090/s0002-9904-1942-07750-0

1088-9485

Reinhold Baer,

Rings, Modules, and Algebras

It may happen that an element in a ring is both a zero-divisor and an inverse, that it possesses a right-inverse though no left-inverse, and that it is neither a zero-divisor nor an inverse.Thus there arises the problem of rinding conditions assuring the absence of these paradoxical phenomena; and it is the object of the present note to show that chain conditions on the ideals serve this purpose.At the same time we obtain criteria for the existence of unit-elements.The following notations shall be used throughout.The element e in the ring R is a left-unit for the element u in R, if eu = u ; and e is a leftunit for R, if it is a left-unit for every element in R. Right-units are defined in a like manner; and an element is a universal unit f or R, if it is both a right-and a left-unit for R.The element u is a right-zero-divisor, if there exists an element v ^ 0 in R such that vu = 0 ; and u is a right-inverse in R, if there exists an element w in R such that wu is a left-unit for u and a right-unit for R. Left-zero-divisors and left-inverses are defined in a like manner.Note that 0 is a zero-divisor, since we assume that the ring R is different from 0.L(u) denotes the set of all the elements x in R which satisfy xu = 0; clearly L(u) is a left-ideal in the ring R and every left-ideal of the form L(u) shall be termed a zero-dividing left-ideal.Principal leftideals 1 are the ideals of the form Rv for v in R and the ideals vR are the principal right-ideals.