# Quick Question on Notation: $R^{\, \oplus n}$

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.)

• It means exactly what you said. – darij grinberg Nov 25 '19 at 0:09
• @darijgrinberg Thanks - some verification was exactly what I needed... – 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. $$\kappa< \infty \implies R^{ \kappa} = R^{\oplus \kappa}.$$