# Algebra....subrings

• Feb 28th 2006, 03:32 PM
suedenation
Algebra....subrings
Give an example of a ring R and a subring S of R so that R and S both have an
identity but 1S is not equal to 1R.

Any idea? I've spent almost an hour to do this, but still haven't got the answer.
The midterm is coming soon, hope someone can help me to solve this problem.

Thanks x 1000 :p
• Feb 28th 2006, 05:23 PM
ThePerfectHacker
Quote:

Originally Posted by suedenation
Give an example of a ring R and a subring S of R so that R and S both have an
identity but 1S is not equal to 1R.

Any idea? I've spent almost an hour to do this, but still haven't got the answer.
The midterm is coming soon, hope someone can help me to solve this problem.

Thanks x 1000 :p

If I understand $\displaystyle 1S$ and $\displaystyle 1R$ as cosests. Then since 1 is identity, $\displaystyle 1S=S$ and $\displaystyle 1R=R$. Thus, the problem reduces to finding a PROPER subring of a ring.
Take any ring $\displaystyle <\mathcal{R},+,\cdot>$ then the trivial subring $\displaystyle \{0\}$ satisfies these conditions.