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

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

ii). Show that the subset I of S = Mn(R) consisting of thosematrices 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 a2 = 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 allidempotent elements of R is closed under multiplication

(iv) A ring B is a Boolean ring if a 2 = a for all a ? B, sothat every element is idempotent. By considering (x + x) 2 showthat 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 areidempotents, then a ? b := a + b ? ab 1 2 PROBLEM SHEET 4 is alsoan idempotent. Show that the set B = Idem(R) of all idempotents ofR is a Boolian ring, where the addition is ? and the multiplicationis the same as in R.

vii) Let E be a set and let B the set of all subsets of E, showthat B is a Booloian ring, where the ”multiplication” of twoelements of B (i.e. subsets of E) is the intersection of thesesubsets, 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 zeroelements of B?