For #1. Is it true that ?
For #2b. If then because the mapping is onto we get .
This means because R is commutative, . Which means
1.
is a subring of a ring (complex)
Find an element of R different from the identity element which has a multiplicative inverse in R.
2. Let R,S be rings and suppose is a ring homomorphism. Suppose
(a) Show that if R has an identity element then S has an identity element.
(b) Show that if R is commutative then S is commutative
(C) Is it true that if R is an integral domain then S is an integral domain? Justify your assertion.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I'm not 100% sure on some of this.
Suppose that there exists an element such that for all . Then we claim that is the identity element for . We need to show for all . Since the function is onto it means . That means and similarly .
No, it is not true. Let and . For let to be the congruence class of mod 4. Then is a ring homomorphism and but is an integral domain and is not.(C) Is it true that if R is an integral domain then S is an integral domain? Justify your assertion.