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

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

环与域代写 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},

1. Let R be a ring and let M_n(R) denote the set of n × n matrices with entries in R. Prove that M_n(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 + b√ 2 : a, b ∈ Z} is a commutative ring under real addition and multiplication.

Find the multiplicative identity of R, and the multiplicative inverses of the elements 1 + √2 and 3 + 2√2.

Show that R is not a field.

9. (1) Prove that the set S = {a + b√ 2 : 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 + · · · + a_nxⁿ 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) = a−bi 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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