189-235A: Basic Algebra I Assignment 4

Due: Wednesday, October 6

1. Let R be a commutative ring. Is the set S of matrices of the form ( \begin{matrix}a & b \\ 0 & c \end{matrix} ),

with a, b, c ∈ R, a subring of M_{2}(R)?

2. Same question as 1, but with S replaced by the set of matrices of the form

( \begin{matrix}a & b \\ 0 & a \end{matrix} ), with a, b ∈ R.

3. Show that the ring Q of rational numbers has no subrings which are finite sets.

4. Let A be a commutative ring, let R = A × A, and let S be the subset of elements of R of the form (a, 0), with a ∈ A. Show that S is a ring which is isomorphic to A. Show that S is not a subring of R. (According to the definition of subring given in class, which differs from the definition given in Hungerford’s book for example.)

5. Let R be a ring, and let a be an element of R which is not a zero divisor. Show that the cancellation law can be applied to a, i.e., for all x, y ∈ R, if ax = ay then x = y.

6. Let R = Q(\sqrt{2}) be the ring of elements of the form a+b\sqrt{2}, with a, b ∈ Q. Show that the function which sends a + b\sqrt{2} to a − b\sqrt{2} is an isomorphism from R to itself.

7. Let R be a ring. Show that there is a unique ring homomorphism f from Z to R.

8. Given a ring R, let R[i] denote the set of pairs (a, b) with (a, b) ∈ R.

Define an addition and multiplication on R[i] by the rules:

(a_{1}, b_{1}) + (a_{2}, b_{2}) = (a_{1} + a_{2}, b_{1} + b_{2}).

(a_{1}, b_{1})(a_{2}, b_{2}) = (a_{1}a_{2} − b_{1}b_{2}, a_{1}b_{2} + b_{1}a_{2}).

Show that these rules equip R[i] with the structure of a ring.

9. Show that the subset S of the ring R[i] of exercise 8 consisting of elements of the form (r, 0) with r ∈ R is a subring of R[i] which is isomorphic to R. Produce an element i in R[i] satisfying i^{2} = −1.

10. Let R be the field of real numbers. Retaining the notations of exercise 8, show that R[i] is isomorphic to the ring C of complex numbers. Optional questions.

11. Keeping the notations of exercise 8, show that the ring C[i] is isomorphic to C × C.

12. Show that a ring R which is a finite set and an integral domain (has no zero-divisors) is necessarily a field.