(a) Give an example showing the following characterisation of a subring is incorrect, and explain how to modify it to make it correct: "A subset S of a ring R is a subring of R if 0 E S and whenever a, b = S we have a+bЄ S and a. be S."

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 ].

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

18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .

Label each of the following statements as either true or false. Every subring of a ring R is an idea of R.

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

a. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R. (Hint: Compare this with Example 4, and note that the element a itself may not be in this set I.) b. Give an example of a commutative ring R and an element aR such that a(a)={ar|rR}.

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

True or False Label each of the following statements as either true or false. If one element in a ring R has a multiplicative inverse, then all elements in R must have multiplicative inverses.

11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .