Right-bounded factors in an LCM domain

Abstract:

A right-bounded factor is an element in a ring that generates a right ideal which contains a nonzero two-sided ideal. Right-bounded factors in an LCM domain are considered as a generalization of the theory of two-sided bounded factors in an atomic $2$-fir, that is, a weak Bezout domain satisfying the acc and dcc for left factors. Although some elementary properties are valid in a more general context most of the main results are obtained for an LCM domain satisfying $({\text {M}})$ and the dcc for left factors; the condition $({\text {M}})$ is imposed to insure that prime factorizations are unique in an appropriate sense. The right bound ${b^ \ast }$ of a right bounded element $b$ is considered in general, then in case $b$ is a prime, and finally in case $b$ is indecomposable. The effect of assuming that right bounds are two-sided is also considered.