MTH6158: Ring Theory Main Examination period 2023

发布时间：2024-05-13

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Main Examination period 2023 – May/June – Semester B

MTH6158: Ring Theory

Question 1 [30 marks].

(a) Give an example of a non-commutative ring without an identity.

(b) id(D)e(o)n(e)ti(s)t(t)y(h)? Expla(e equati)in(on) (1 + a)(1 − a) = 1 − a2 hold for any element a of a ring with

(d) Prove, using the axioms of a ring or the basic properties proved in the lectures, that any two elements a,b of a ring satisfy the equation (−a)b = − (ab).

(e) Give an example of a commutative ring without identity having a subring with identity, or explain why such an example cannot exist.

(f) Explain what is wrong in the following “proof” that every finite commutative ring with identity is a field.

“Proof”: Suppose R is a finite commutative ring with identity. Let a be a

non-zero element of R. We want to show that there exists an inverse of a in R, that is, an element b such that ab = ba = 1. Consider the set S = {a,a2 , a3 , . . . }.

Since R is finite, this set S must be finite. This means that there exist positive

integers m > n such that am = an. We then have am −n = 1, which means that the element am −n − 1 is a multiplicative inverse of a. Thus every non-zero element of R has an inverse, and therefore R is a field.

Question 2 [20 marks]. Consider the ring R = Z/15Z and its ideal

I = {[0]15 , [3]15 , [6]15 , [9]15 , [12]15 }. [You are not required to prove that I is an ideal of R.]

(a) Is the ideal I a ring with identity? Explain.

(b) Write down explicitly the partition of R into cosets of I.

(d) Does the equation x3 + x5 + x7 = 1 have a solution in the ring R/I? Explain.

Question 3 [30 marks].

(a) Give an example of a domain R and an element a ∈ R that is neither a unit nor a zero-divisor.

(b) For which integers m ≥ 2 does the ring Z/mZ satisfy the cancellative law for multiplication? Explain.

(i) Explain why S is an integral domain.

(ii) Show that the element 2 + √3 is a unit of S.

(iii) Find a factorisation of the element 6 ∈ S as a product of two elements of S that are not in Z.

(iv) Given that the element 6 ∈ S can also be factored as 6 = 2 · 3, can we conclude that S is not a unique factorisation domain? Explain.

(d) Suppose R is a domain and a ∈ R is a non-zero element satisfying a3 = a. Show that a is either a unit or a zero-divisor.

Question 4 [20 marks]. Consider the field of 2 elements K = Z/2Z and the

(a) Explain why f is an irreducible element of K[x].

(b) Let F be the quotient ring F = K[x]/⟨f⟩, which contains the field K.

(i) Explain why F is a field. [You may use any result proved in the lectures.]

(ii) How many elements does the field F have?

(iii) Let α be an element of F such that f(α) = 0. Find an expression for the inverse α− 1 of the form α− 1 = a · α2 + b · α + c with a,b,c ∈ K.