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<p>We define and examine in-depth the notion of a <i>weakly reversible ring</i> showing that all weakly reversible rings are abelian McCoy rings and so, in particular, they are abelian 2-primal rings. Moreover, we construct a weakly reversible ring which is <i>not</i> reversible. We also prove that, if <i>R</i> is a weakly reversible ring, then the polynomial ring <i>R</i>[<i>x</i>] is strongly AB. Thus, in particular, the weakly reversible ring <i>R</i> is zip if, and only if, <i>R</i>[<i>x</i>] is zip. We, moreover, establish that if <i>R</i> is a weakly reversible ring and every prime ideal of <i>R</i> is maximal, then both <i>R</i> and <i>R</i>[<i>x</i>] are AB rings.</p>

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A new class of rings having the McCoy condition

  • 期刊论文

  • 发表日期：

    2026年7月23日

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  • Peter Danchev,
  • Masoome Zahiri

摘要

We define and examine in-depth the notion of a weakly reversible ring showing that all weakly reversible rings are abelian McCoy rings and so, in particular, they are abelian 2-primal rings. Moreover, we construct a weakly reversible ring which is not reversible. We also prove that, if R is a weakly reversible ring, then the polynomial ring R[x] is strongly AB. Thus, in particular, the weakly reversible ring R is zip if, and only if, R[x] is zip. We, moreover, establish that if R is a weakly reversible ring and every prime ideal of R is maximal, then both R and R[x] are AB rings.