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环与域代写 MT3505Algebra代写 Algebra代写 Rings & Fields代写

419

MT3505 Algebra: Rings & Fields: Solutions for Chapter 1.

1. Let R be a ring and let Mn(R) denote the set of n × n matrices with entries in R. Prove that Mn(R) is a ring.

2. For an integer n N \ {0}, the set of congruence classes of the integers modulo n forms a ring, called the ring of integers modulo n. Addition and multiplication are given by: for a, b Z,

(1) Complete the proof from lectures that these operations are well-defined and that they make Z/nZ into a ring.

(2) Write out the addition and multiplication tables for the ring Z/5Z. Which of the 10 possible axioms does Z/5Z satisfy? Is it a field?

(3) Now, write out the addition and multiplication tables for the ring Z/6Z. Which of the 10 possible axioms does it satisfy? Is it a field?

3.

Is M a commutative ring?

Does M have an identity?

4. Let R be any ring. Use the ring axioms to prove that

(1) (a) = a;

(2) (a) b = a (b) = (a b);

(3) (a) (b) = a b;

Let R have a multiplicative identity 1. Prove that

(4) (1) a = a;

(5) (1) (1) = 1;

(6) if 1 = 0 in R, then |R| = 1. If 0 = 1, then

5. 环与域代写

Let R be a set with operations + and satisfying the ring axioms except possibly A1 (commutativity of addition). Prove that if R satisfies M3 (existence of a unity element) then A1 holds.

6. Let G be an (additive) abelian group. Prove that, if we define an operation of multiplication in G by

a b = 0 for all a, b G

then G is a ring.

7. Let R = R × R = {(a, b) : a, b R}. Define + and by

(a, b) + (c, d) = (a + c, b + d)

(a, b)(c, d) = (ac, bd).

Prove that R is a commutative ring with identity.

Find the elements of R that have a multiplicative inverse.

8. Prove that the set R = {a + b2 : a, b Z} is a commutative ring under real addition and multiplication.

Find the multiplicative identity of R, and the multiplicative inverses of the elements 1 + 2 and 3 + 22.

Show that R is not a field.

9. (1) Prove that the set S = {a + b2 : a, b Q} is a field under real addition and multiplication.

(2) Let E be a subfield of R containing 2. Show that S E (this shows that S is the “smallest” (under containment) subfield of R which contains 2).

11. 环与域代写

Let R be a ring where x2 = x ∗ x = x for all x ∈ R. Prove that, in R, the following hold: (i) 2x = x + x = 0 for all x ∈ R; (ii) R is commutative

Prove that H is not an integral domain.

13. Let R be a ring. Prove that R[x] is a ring, with zero element the zero polynomial.

Prove that R[x] is commutative if and only if R is commutative.

14. Let R be a ring and let f, g R[x]. By using the formulae for addition and multiplication, prove that

(i) deg(f + g) max(deg(f), deg(g)) and deg(f g) deg(f) + deg(g).

(ii) If R is an integral domain, then deg(f g) = deg(f) + deg(g).

(iii) Show that, even when R is an integral domain, it is not the case that deg(f + g) = max(deg(f), deg(g)) for all f, g ∈ R[x].

15. 环与域代写

Determine which of the following are subrings of the given rings.

A non-empty subset A of a ring R is a subring if and only if a b A and a b A for all a, b A.

(1) the positive integers in Z;

(2) all polynomials with integer constant in Q[x];

(3) all polynomials of degree at least 6 in Q[x];

(4) the set of polynomials a0 + a1x + a2x2 + · · · + anxn in R[x] for which a0 = a1 = 0;

16. (1) Prove that the centre of a ring R is a subring of R.

(2) Let R = M2(R). Prove that the centre of R is

17. Let f : R → S and g : S → T be ring homomorphisms. Prove that g ◦ f : R → T is a ring homomorphism.

19. Let f(a+bi) = abi denote the conjugation map on the complex numbers C. Prove that f is a homomorphism from Z[i] to Z[i].

20. Which of the maps from C to C given by the following are ring homomorphisms

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