Ring theory convention

Question

Given $0$ in the integers, and a member $r$ of a ring $R$, is it a convention that the following is satisfied:

$0_{\mathbb{Z}}r=0_R$?

Note ($r+r+r+...+r$ (n times) is defined to be $nr_1$, where $n$ is an integer)

Answer 1

We define $nr=\underbrace{r+r+\cdots+r}_{n\text{ times}}\in R$ for $n\in\mathbb Z_{>0}$, evidently $(n+m)r=nr+mr$ for $m,n\in\mathbb Z_{>0}$. If we want this distributive law to still hold true for arbitrary $m,n\in\mathbb Z$, which we absolutely do, then we are forced to accept $mr=(m+0_{\mathbb Z})r=mr+0_{\mathbb Z}r$ and hence that $0_{\mathbb Z}r=0_R$, since there is only one additive identity in a ring. So the statement $0_{\mathbb Z}r=0_R$ is true, pretty much by definition.

Answer 2

In complement to YiFan's answer, there is a convention that if we perform a sum zero times the answer is zero in whatever ring we are working in. Similarly, in an iterated product we perform zero times the answer will be $1_R$ if such an identity exists. These conventions make a lot of degenerate cases make sense in such formulas.