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In the paper titled:

On the module of differentials of a noncommutative algebra and symmetric biderivations of a semiprime algebra

I found the following definition:

Let $k$ be a commutative ring with an identity element, let $R$ be a $k$-algebra (not necessary with an identity element). An additive abelian group $M$ is called a left $R/k$-module if it is a left R-module and a unitary $k$-module satisfying $a(xm) = (ax)m = x(am)$ for all $m \in M$, $x \in R$ and $a \in k$.

I search a reference (i.e. book) where I can find the definition of this kind of modules.

Thanks.

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1 Answer 1

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This is a very basic question. See e.g. Section 1.1 of the book:

R.S. Pierce: Associative algebras, Graduate Texts in Mathematics, 88. Springer-Verlag, 1982.

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  • $\begingroup$ Thanks, but I did not find the definition in the book you mentioned (1.1. Conventions subsection), I asked for modules denoted by this notation -- $R/k$-module --. $\endgroup$
    – The Student
    May 16, 2022 at 14:50
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    $\begingroup$ Although the notation is different, this is just the notion you are looking for. As a matter of fact, in the mentioned book the author assumes that $A$ is unital, but the definition naturally extends to the non-unital case as well. Look also at the following questions on StackExchange: math.stackexchange.com/questions/211602/… and math.stackexchange.com/questions/1516990/… $\endgroup$
    – Salvatore Siciliano
    May 16, 2022 at 15:38

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