Rings, subrings problem

• Apr 19th 2013, 07:58 PM
christianwos
Rings, subrings problem
Recall that an identity for a ring R is an element 1_R in R such that for each for each r in R,
1_R*r = r = r*1_R
(a) Show that there is a ring R with identity 1_R and a subring S of R not containing 1_R,
but such that S has its own identity 1_S not equal to 1_R.
(b) Show that if R is an integral domain then for every subring S with identity 1_S, 1_S = 1_R.

Your help would be much appreciated. Thank you.
• Apr 21st 2013, 04:39 PM
Ant
Re: Rings, subrings problem
Quote:

Originally Posted by christianwos
Recall that an identity for a ring R is an element 1_R in R such that for each for each r in R,
1_R*r = r = r*1_R
(a) Show that there is a ring R with identity 1_R and a subring S of R not containing 1_R,
but such that S has its own identity 1_S not equal to 1_R.
(b) Show that if R is an integral domain then for every subring S with identity 1_S, 1_S = 1_R.

Your help would be much appreciated. Thank you.

For (a) consider $\mathbb{Z} \oplus \mathbb{Z}$ then is $(1,1)$ is the identity. Consider the subring $\{(a,0) : a \in \mathbb{Z} \}$ .....