Let R be an abelian group. View R as a ring by declaring the multiplication operation is ab = 0 for all a, b = R. Prove that every subgroup S ≤ R is actually an ideal.

14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.

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.

21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.

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

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

An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.

24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)

Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4

a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].