In Rings and Modules theory, for a ring (or a field or module), $R$, what does the notation $R^{\, \oplus n}$ denote? (Here $n \in \mathbb{N}$.)

I assume that it just means

$$ R^{\, \oplus n} = \underbrace{R \oplus R \oplus \cdots \oplus R}_\text{n times} $$

However, I wanted to check that this is not standard notation for something else that I am not aware of.

In the question in which I encountered this notation, both the notation $R \oplus R$ and $R^{\, \oplus 2}$ is used on consecutive lines, leading me to worry that they might denote different things. (This notation is not referenced or explained anywhere else in the question set / lecture notes etc.)

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    $\begingroup$ It means exactly what you said. $\endgroup$ – darij grinberg Nov 25 '19 at 0:09
  • $\begingroup$ @darijgrinberg Thanks - some verification was exactly what I needed... $\endgroup$ – bobstalobsta Nov 25 '19 at 0:12

The notation $$R^{\oplus \kappa}\overset{\mathrm{def}}{:=}\{f \in \mathrm{Hom}(\kappa,R) \ | \ \#|\kappa \setminus f^{-1}(0)| < \infty \}$$ is sometimes used to distinguish it from $$R^{ \kappa} \overset{\mathrm{def}}{:=} \mathrm{Hom}(\kappa,R)$$ when $\kappa$ an infinite cardinal and $R$ is any general algebraic structure. In other words $$R^{\oplus \kappa} = \bigoplus_{i \in \kappa} R$$ and $$R^{ \kappa} = \prod_{i \in \kappa} R.$$ In the finite case these two are the same thing, i.e. $$\kappa< \infty \implies R^{ \kappa} = R^{\oplus \kappa}.$$


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