1. ## 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.

2. ## Re: Rings, subrings problem

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