Consider the ring R = 3Z and the ideal I = 12Z of R. Use the Second Isomorphism Theorem to list all the subrings of R/I.

Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.

22. Let be a ring with finite number of elements. Show that the characteristic of divides .

Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)

Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an ideal of R.

15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .

17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.

If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.

Assume that the set R={[x0y0]|x,y} is a ring with respect to matrix addition and multiplication. Verify that the mapping :R defined by ([x0y0])=x is an epimorphism from R to Z. Describe ker and exhibit an isomorphism from R/ker to

Let :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove that is a ring homomorphism. Is (e)=e where e is the unity in 3 and e is the unity in 12?