Rings whose Non-Units are a Unit Multiple of an Element from $\sqrt{Δ(R)}$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912907100422144 |
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| author | Hasanzadeh, Omid Moussavi, Ahmad Danchev, Peter |
| author_facet | Hasanzadeh, Omid Moussavi, Ahmad Danchev, Peter |
| contents | This paper introduces and studies a new class of rings called {\it $U\sqrtΔ$-rings}. A ring $R$ is $U\sqrtΔ$ if every non-unit element can be written as the product of a unit and an element from $\sqrt{Δ(R)}$, where $\sqrt{Δ(R)}$ consists of elements some power of which lies in the special subring $Δ(R)$. We establish certain basic properties of these rings and, concretely, prove that they are simultaneously indecomposable and Dedekind-finite. We also show that the polynomial ring $R[x]$ and the Laurent polynomial ring $R[x, x^{-1}]$ are never $U\sqrtΔ$-rings, while the power series ring $R[[x]]$ inherits this property from $R$. Likewise, for left (right) Artinian rings, the conditions of being a $U\sqrtΔ$-ring and a $UN$-ring are equivalent, as well as these two conditions are preserved for the full matrix ring $M_n(R)$ of size $n\geq 1$ over $R$. In addition, for a commutative ring $R$, $M_n(R)$ is a $U\sqrtΔ$-ring exactly when $R$ is local. Furthermore, we characterize when a group ring $RG$ is a $U\sqrtΔ$-ring showing that, for a locally solvable group $G$, this occurs precisely when $R$ is a $U\sqrtΔ$-ring and $G$ is a locally finite $p$-group for some prime $p \in J(R)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14600 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rings whose Non-Units are a Unit Multiple of an Element from $\sqrt{Δ(R)}$ Hasanzadeh, Omid Moussavi, Ahmad Danchev, Peter Rings and Algebras 16N20, 16S34, 16S50, 16U99 This paper introduces and studies a new class of rings called {\it $U\sqrtΔ$-rings}. A ring $R$ is $U\sqrtΔ$ if every non-unit element can be written as the product of a unit and an element from $\sqrt{Δ(R)}$, where $\sqrt{Δ(R)}$ consists of elements some power of which lies in the special subring $Δ(R)$. We establish certain basic properties of these rings and, concretely, prove that they are simultaneously indecomposable and Dedekind-finite. We also show that the polynomial ring $R[x]$ and the Laurent polynomial ring $R[x, x^{-1}]$ are never $U\sqrtΔ$-rings, while the power series ring $R[[x]]$ inherits this property from $R$. Likewise, for left (right) Artinian rings, the conditions of being a $U\sqrtΔ$-ring and a $UN$-ring are equivalent, as well as these two conditions are preserved for the full matrix ring $M_n(R)$ of size $n\geq 1$ over $R$. In addition, for a commutative ring $R$, $M_n(R)$ is a $U\sqrtΔ$-ring exactly when $R$ is local. Furthermore, we characterize when a group ring $RG$ is a $U\sqrtΔ$-ring showing that, for a locally solvable group $G$, this occurs precisely when $R$ is a $U\sqrtΔ$-ring and $G$ is a locally finite $p$-group for some prime $p \in J(R)$. |
| title | Rings whose Non-Units are a Unit Multiple of an Element from $\sqrt{Δ(R)}$ |
| topic | Rings and Algebras 16N20, 16S34, 16S50, 16U99 |
| url | https://arxiv.org/abs/2602.14600 |