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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2405.08364 |
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| _version_ | 1866916340353204224 |
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| author | Wehrung, Friedrich |
| author_facet | Wehrung, Friedrich |
| contents | A map $f\colon R\to S$ between (associative, unital, but not necessarily commutative) rings is a\emph{brachymorphism} if $f(1+x)=1+f(x)$ and $f(xy)=f(x)f(y)$ whenever $x,y\in R$. We tackle the problem whether every brachymorphism is additive (i.e., $f(x+y)=f(x)+f(y)$), showing that in many contexts, including the following, the answer is positive: $R$ is finite (or, more generally, $R$ is left or right Artinian); $R$ is any ring of $2\times2$ matrices over a commutative ring; $R$ is Engelian; every element of $R$ is a sum of $π$-regular and central elements (this applies to $π$-regular rings, Banach algebras, and power series rings); $R$ is the full matrix ring of order greater than $1$ over any ring; $R$ is the monoid ring $K[M]$ for a commutative ring $K$ and a $π$-regular monoid $M$; $R$ is the Weyl algebra $A_1(K)$ over a commutative ring $K$ with positive characteristic; $f$ is the power function $x\mapsto x^n$ over any ring; $f$ is the determinant function over any ring $R$ of $n\times n$ matrices, with $n\geq3$, over a commutative ring, such that if $n>3$ then $R$ contains $n$ scalar matrices with non zero divisor differences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Is addition definable from multiplication and successor? Wehrung, Friedrich Rings and Algebras A map $f\colon R\to S$ between (associative, unital, but not necessarily commutative) rings is a\emph{brachymorphism} if $f(1+x)=1+f(x)$ and $f(xy)=f(x)f(y)$ whenever $x,y\in R$. We tackle the problem whether every brachymorphism is additive (i.e., $f(x+y)=f(x)+f(y)$), showing that in many contexts, including the following, the answer is positive: $R$ is finite (or, more generally, $R$ is left or right Artinian); $R$ is any ring of $2\times2$ matrices over a commutative ring; $R$ is Engelian; every element of $R$ is a sum of $π$-regular and central elements (this applies to $π$-regular rings, Banach algebras, and power series rings); $R$ is the full matrix ring of order greater than $1$ over any ring; $R$ is the monoid ring $K[M]$ for a commutative ring $K$ and a $π$-regular monoid $M$; $R$ is the Weyl algebra $A_1(K)$ over a commutative ring $K$ with positive characteristic; $f$ is the power function $x\mapsto x^n$ over any ring; $f$ is the determinant function over any ring $R$ of $n\times n$ matrices, with $n\geq3$, over a commutative ring, such that if $n>3$ then $R$ contains $n$ scalar matrices with non zero divisor differences. |
| title | Is addition definable from multiplication and successor? |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2405.08364 |