Question

Let R be a ring and n ∈ N. Let S = Mn(R) be the ring of n × n matrices with entries in R.

a) i) Let T be the subset of S consisting of the n × n diagonal matrices with entries in R (so that T consists of the matrices in S whose entries off the leading diagonal are zero). Show that T is a subring of S. We denote the ring T by Dn(R).

ii). Show that the subset I of S = Mn(R) consisting of those matrices A = (aij ) ∈ Mn(R) with aij = 0 whenever 1 ≤ i ≤ n and 1 ≤ j ≤ n − 1 is a left ideal of Mn(R).

b) Let R be a ring. An element a in R is called idempotent if a 2 = a.

i) Find all idempotent elements of the rings R = Z/6Z, and S = Z[i]

(ii) Show that if a is an idempotent in a ring R, then so is b = 1 − a.

(iii) Show that if R is a commutative ring, then the set of all idempotent elements of R is closed under multiplication

(iv) A ring B is a Boolean ring if a 2 = a for all a ∈ B, so that every element is idempotent. By considering (x + x) 2 show that 2a = 0 for any element a in a Boolean ring B.

v) Show that if B is a Boolean ring, then B is commutative.

vi) Show that if R is a commutative ring and a and b are idempotents, then a ⊕ b := a + b − ab 1 2 PROBLEM SHEET 4 is also an idempotent. Show that the set B = Idem(R) of all idempotents of R is a Boolian ring, where the addition is ⊕ and the multiplication is the same as in R.

vii) Let E be a set and let B the set of all subsets of E, show that B is a Booloian ring, where the ”multiplication” of two elements of B (i.e. subsets of E) is the intersection of these subsets, while the addition in B is given by X + Y = (X ∪ Y ) \ (X ∩ Y ) Here X, Y ∈ B (so, X ⊂ E, Y ⊂ E). What are the unit and zero elements of B?

Answer 1