Rings in which every nonzero weakly prime ideal is prime

In this paper, we study WP-ring, that is a ring in which every nonzero weakly prime ideal is prime. Some results including the characterizations and the transfer of WP-ring property to homomorphic image and localization are given. Also, we study the possible transfer of the WP-ring property between A and A∝E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\propto E$$\end{document} and between A and A⋈fJ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\bowtie ^{f}J$$\end{document}. Our results provide new classes of commutative rings satisfying the above property.