The McCoy Condition on Ore Extensions

Abstract

Nielsen [29] proved that all reversible rings are McCoy and gave an example of a semicommutative ring that is not right McCoy. When R is a reversible ring with an (α, δ)-condition, namely (α, δ)-compatibility, we observe that R satisfies a McCoy-type property, in the context of Ore extension R[x; α, δ], and provide rich classes of reversible (semicommutative) (α, δ)-compatible rings. It is also shown that semicommutative α-compatible rings are linearly α-skew McCoy and that linearly α-skew McCoy rings are Dedekind finite. Moreover, several extensions of skew McCoy rings and the zip property of these rings are studied.

Key Words:

• McCoy ring
• Reversible ring
• Semicommutative ring
• Zip ring

2010 Mathematics Subject Classification:

• Primary: 16U80
• Secondary: 16S36

ACKNOWLEDGMENT

The authors would like to express their deep gratitude to the referee for a very careful reading of the article, and many valuable comments, which have greatly improved presentation of the article.

Notes

Communicated by V. A. Artamonov.