On LPI rings

Abstract

In this paper, we extend the $LPI$ property (that is, every locally principal ideal in an integral domain is invertible) to rings with zero-divisors and we study the class of commutative rings in which every regular locally principal ideal is invertible called $LPI$ rings. We investigate the stability of this property under homomorphic image, and its transfer to various contexts of constructions such as direct products, amalgamation of rings and trivial ring extensions. Our results generate examples which enrich the current literature with new and original families of rings that satisfy this property.

Keywords

• Locally principal ideals,regular ideals,trivial extension,pullback,amalgamation of rings,LPI-rings

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