RINGS AND FIELDS ASSIGNMENT THREE 21 MAY 2022 Due Monday 6th June 2022 1. (a) Let R be a ring with identity element 1R and show that the distributive laws imply that x +y = y + x for all x, y € R. (Hint: expand (x + y)(1R+ 1R) in two different ways.) [2] (b) Let R be a ring with identity element 18 and explain why a proper ideal of R cannot contain a unit. [2] 2. Let M₂(Z) be the ring of 2x2 matrices whose entries are integers. (a) Find any two matrices in R which are zero-divisors; [2] (b) Find a subring of R which is not an ideal of R; [3] (c) Find any infinite subset of the group of units of R (i.e. describe an infinite subset of M₂(Z) whose elements are units). [4] 3. Let S be a non-empty subset of a commutative ring R and define Ann(S) = {r € R: rx = OR for all x € S} Show that Ann(S) is non-empty (give an example of at least one element in Ann(S)) and prove that Ann(S) is an ideal of R. (This is known as the annihilator of S in R). [4] 4. Let R be a ring which satisfies the equation x² = x for all x = R and show that (a) x = -x for all x € R; [2] (b) xy = yx for all x and y in R (Hint: calculate (x+y)² and use (a)) [3] (c) if R is an integral domain what would be the characteristic of R. Give reasons for your answer. (R is called a Boolean ring) [2] 5. Let R=Z7, the filed with 7 elements and let f(x) = x³+x²+x+2 be an element of the polynomial ring Z-[x]. Prove that (a) f(x) is an irreducible element of the principal ideal domain R. (Hint: if f(x) = g(x)h(x) then either degree(g(x)) = 1 or degree(h(x)) = 1 which means that f(x) has a root in R. Show directly that this is not the case); [3] (b) the field F = R[x]/ has 7³ elements. (Hint: if h(x) + is an arbitrary element of F, then by Division Algorithm, h(x) + =r(x) + , where degree(r(x)) <3). [3]