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Let $K$ be a commutative ring with unit and suppose that $R$ is a ring that is a left $K$-module satisfying $c(rs)=(cr)s$ for all $c\in K$ and $r,s\in R$. We do not require that $r(cs)=c(rs)$ and so $R$ need not be a $K$-algebra. I also don't require $R$ to be unital and this is important. Is there a name for such a structure?

A typical example is if $K$ is a commutative ring, $S$ is a semigroup acting by endomorphisms and $c\colon S\times S\to K^{\times}$ is a $2$-cocycle, then the crossed product ring $R$ is a left $K$-module satisfying the above property but is not a $K$-algebra unless $S$ acts trivially on $K$. If $S$ is not a monoid this can fail to be unital.

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