On weakly (1,n)-ideals and weakly n-ideals

Abstract

We study weakly (1, n)-ideals and weakly n-ideals in commutative rings. Let A be a commutative ring with a nonzero identity and I be a proper ideal of A. Then I is said to be a weakly (1, n)-ideal (or weakly n-ideal) if whenever 0 not equal abc is an element of I for some nonunits a,b,c is an element of A (or 0 not equal ab is an element of I for some a,b is an element of A), then either ab is an element of I or c is an element of N(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c \in \mathfrak{N}(A)$$\end{document} (or a is an element of I or b is an element of N(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b \in \mathfrak{N}(A)$$\end{document}, respectively), where N(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{N}(A)$$\end{document} is the set of all nilpotent elements of A. Many examples and properties of weakly (1, n)-ideals and weakly n-ideals are given. We characterize all rings in which every proper ideal is a weakly (1, n)-ideal and weakly n-ideal. Furthermore, we investigate both weakly (1, n)-ideals and weakly n-ideals in amalgamated algebras along an ideal.