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Right ${\rm LCM}$ domains

Raymond A. Beauregard

1971
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Proceedings of the American Mathematical Society
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A right LCM domain is a not necessarily commutative integral domain with unity in which the intersection of any two principal right ideals is again principal. The principal result deals with right LCM domains that satisfy an additional mild hypothesis; for such rings (which include right HCF domains and weak Bezout domains) it is shown that each prime factorization of an element is unique up to order of factors and projective factors. Projectivity is an equivalence relation that reduces to the
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... hat reduces to the relation of "being associates" in commutative rings and reduces to similarity in weak Bezout domains. All rings considered are (not necessarily commutative) integral domains with unity. If R is such a ring, R* denotes the monoid of nonzero elements of R. Among the interesting properties of integral domains are the conditions that guarantee uniqueness of the prime decompositions of a given element. (A prime is understood to be a nonzero nonunit with no proper divisors.) In commutative rings the uniqueness referred to is uniqueness up to order of factors and associate factors. It is common knowledge that, in this case, uniqueness is guaranteed by the existence of LCM's for each pair of nonzero elements. This is, of course, equivalent to existence of GCD's for such elements, although in the noncommutative case this is not true. Our main concern is the extension of such results to the noncommutative case. In going over to the noncommutative case uniqueness must be weakened. It is well known [2] that the prime factorization of a given element is unique up to order of factors and similarity in a weak Bezout domain, that is, in a ring in which the sum and intersection of any two principal right (or left) ideals is again principal whenever the intersection is nonzero. Now a weak Bezout domain can be characterized as a ring in which right (and left) LCM's exist for each pair of nonzero elements that has a nonzero common right (left) multiple, and left (and right) GCD's exist for such elements and are Received by the editors August 17, 1970. A MS 1969 subject classifications. Primary 1615.

doi:10.1090/s0002-9939-1971-0279125-1
fatcat:awxkff2jw5cjthgzm3bu7imy2a