## Dr. Sarah's Problem Set 7 on Rings, Fields and Integral Domains

### If the statement is true, then
give an example and informal justifications (no formal
proof necessary, but your informal justifications need to be
clear and complete, similar to WebCT postings).
If the statement is false, then prove that
it is false (ie write down the negation and prove it).

1) There exists a finite ring which is not abelian under multiplication.
2) There exists an infinite ring which is not abelian under multiplication.
3) There exists a group G, there exists a,b not 0 in G so that
ax=b has more than one solution.
4) There exists a ring R, there exists a,b not 0 in R so that
ax=b has more than one solution.
5) There exists a ring
that is abelian under *, has no 0 divisors, but is not
an integral domain.
6) There exists a ring R, there exists a,b in R so that
ab=0 but ba is not 0.
7) There exists a ring R, there exists a subset S of R so that
S is a subgroup under addition, but S is not a subring.
8) There exists a ring R so that
R is a finite integral domain and R is not a field.
9) I1 and I2 integral domains ---> I1 direct sum I2 is an integral domain
(recall that the direct sum of two rings is a ring).
10) For all p(x) in Z_8[x] (polynomials in x with coefficients in Zmod8)
p(x)*(4x^2+6x+3) is not equal to 1 mod 8.
### Addition and Multiplication Mod 10

Prove that Z_10 is a ring (you may assume associative and distributive
properties hold).
Prove that Z_10 is not a field.
Prove that Z_10 is not an integral domain.
Prove that {0,2,4,6,8} under +mod 10 and multiplication mod 10
is a field.

**For proofs, be sure to use proofwriting format for full
credit!**