Skew Generalized Power Series Rings With the McCoy Property
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| Format: | Preprint |
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2025
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| _version_ | 1866908339993051136 |
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| author | Danchev, Peter Zahiri, M. Zahiri, S. |
| author_facet | Danchev, Peter Zahiri, M. Zahiri, S. |
| contents | Let $R$ be a ring, $(S,\preceq)$ a strictly totally ordered monoid and suppose also $ω:S\rightarrow \text{End}(R)$ is a monoid homomorphism. A skew generalized power series ring $R[[S,ω,\preceq]]$ consists of all functions from a monoid $S$ to a coefficient ring $R$ whose support contains neither infinite descending chains nor infinite anti-chains, equipped with point-wise addition and with multiplication given by convolution twisted by an action $ω$ of the monoid $S$ on the ring $R$.
Special cases of the skew generalized power series ring construction are the skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Malcev-Neumann series rings and generalized power series rings as well as the untwisted versions of all of these objects.
In the present article, we study the so-termed $(S,ω)$-McCoy condition on $R$, that is a generalization of the standard McCoy condition from polynomials to skew generalized power series, thus generalizing some of the existing results in the literature relevant to the subject. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_19241 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Skew Generalized Power Series Rings With the McCoy Property Danchev, Peter Zahiri, M. Zahiri, S. Rings and Algebras Representation Theory 16D15, 16D40, 16D70 Let $R$ be a ring, $(S,\preceq)$ a strictly totally ordered monoid and suppose also $ω:S\rightarrow \text{End}(R)$ is a monoid homomorphism. A skew generalized power series ring $R[[S,ω,\preceq]]$ consists of all functions from a monoid $S$ to a coefficient ring $R$ whose support contains neither infinite descending chains nor infinite anti-chains, equipped with point-wise addition and with multiplication given by convolution twisted by an action $ω$ of the monoid $S$ on the ring $R$. Special cases of the skew generalized power series ring construction are the skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Malcev-Neumann series rings and generalized power series rings as well as the untwisted versions of all of these objects. In the present article, we study the so-termed $(S,ω)$-McCoy condition on $R$, that is a generalization of the standard McCoy condition from polynomials to skew generalized power series, thus generalizing some of the existing results in the literature relevant to the subject. |
| title | Skew Generalized Power Series Rings With the McCoy Property |
| topic | Rings and Algebras Representation Theory 16D15, 16D40, 16D70 |
| url | https://arxiv.org/abs/2504.19241 |