. 1. M(R) := {a ER | ax = xa, for all x E R} is called the center of the ring R. Show that (0) M(R)

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1 M R A Er Ax Xa For All X E R Is Called The Center Of The Ring R Show That 0 M R R Ii M R R R I 1 (45.04 KiB) Viewed 27 times

. 1. M(R) := {a ER | ax = xa, for all x E R} is called the center of the ring R. Show that (0) M(R) <R (ii) M(R) = R R is a commutative ring. 2. Let R be a ring. An element a ER is called an idempotent element if a= a. A ring R is called a Boolean ring if every element of Ris idempotent. Show that (i) Every Boolean ring is commutative. (ii) Z is not a Boolean ring. (The only idempotents are 0 and 1.) (iii) Z2 is a Boolean ring. (iv) Zx Z is not a Boolean ring. (The only idempotents are (0,0), (0,1),(1,0) and (1,1).) 3. Let R be a ring. An element a E R is called a nilpotent element if a" = OR for some positive integer n. Show that if a nonzero element a E R is idempotent, then it is not a nilpotent.