USING IDEALS TO PROVIDE A UNIFIED APPROACH TO UNIQUELY CLEAN RINGS

作者：Hiremath V A; Hegde Sharad^*

来源：Journal of the Australian Mathematical Society, 2014, 96(2): 258-274.

DOI：10.1017/S1446788713000591

• semipotent ring
• uniquely clean ring
• ideal extension

In this article, we introduce the notion of the uniquely I-clean ring and show that, if R is a ring and I is an ideal of R then R is uniquely I-clean if and only if (R/I is Boolean and idempotents lift uniquely modulo I) if and only if (for each a is an element of R there exists a central idempotent e is an element of R such that e - a is an element of I and I is idempotent-free). We examine when ideal extension is uniquely clean relative to an ideal. Also we obtain conditions on a ring R and an ideal I of R under which uniquely I-clean rings coincide with uniquely clean rings. Further we prove that a ring R is uniquely nil-clean if and only if (N(R) is an ideal of R and R is uniquely N(R)-clean) if and only if R is both uniquely clean and nil-clean if and only if (R is an abelian exchange ring with J(R) nil and every quasiregular element is uniquely clean). We also show that R is a uniquely clean ring such that every prime ideal of R is maximal if and only if R is uniquely nil-clean ring and N(R) = Nil(*)(R).

• 出版日期2014-4