If $1_R \in M$, an ideal, $M=R $

Question

Let $R$ be a commutative ring with $1_R$ and M be an ideal.

If $1\in M$, $M = $R?

@Randall isn't ideal for the addition? why is always so $r=1\cdot r$? — Beverlie, Jun 05 '18 at 00:59
No, an ideal is a subring closed under internal-external products (loosely speaking). — Randall, Jun 05 '18 at 01:00

score 1 · Accepted Answer · answered Jun 05 '18 at 01:05

1

It is enough to prove that $R \subseteq M$, that is, if $r \in R$, then $r \in M$.

Let $r \in R$. Then $r = r \cdot 1 \in R \cdot M \subseteq M$.

answered Jun 05 '18 at 01:05

lhf

1 Answers1