Question

Let$S$ be a subring of a ring$R$ . Prove that $0_S=0_R$.

Short answer

Hence it is proved that $0_R=0_S$.

Step-by-step solution

Property of Rings: 

If any ringis designated as$R$,such that $a,b\in R$, then:

$a=b\iff a-b=0$

Proof:

It is given that $S$is a subring of a ring $R$. Let there be elements such that:

$a,b\in R$

Then, according to the definition, we have:

role="math" localid="1646821500157" $a+x=b$

This equation will have a unique solution.

Then, for $a\in S$, the below equation will be true, which is given by:

$a+x=a$

Since $S$ is a subring of a ring $R$, therefore, we have:

$x=0_R=0_S$

Hence it is proved that $0_R=0_S$.