I need to prove only number 6. this is about Subrings, Extensions, and Direct Su

Question

I need to prove only number 6. this is about Subrings, Extensions, and Direct Sums in Abstract Math.

Let S and T be subrings of a ring R. For each of the questions below, give a proof or a pair of example Is S union T always, sometimes, or never a subring of R Is S subset T always, sometimes, or never a subrings of R Is S delta T always, sometimes, or never a subring of R Is every subring of a commutative ring necessarily commutative Give a proof or counterexample to justify your answer. Let R be a ring, and let r element of R. The centralizer of r is defined to be the set of all x element of R such that xr = rx. Prove that the centralizer of r is a subring of R. Is the centralizer of r necessarily commutative Explain Let R be a ring, and let r element of R. Prove that S = {x element of R: rx = 0} is a subring of R. Let R be a ring, and let n element of Z. Prove that S = {x element of R: nx = 0} is a subring of R. Let R be a ring with identity, and let S be the set of all units of R. Is S always, sometimes, or never a subring of R Give a proof or a pair of example to justify your answer.

Explanation / Answer

Denote centralizer of R by C

a)

1.

For all x in R

1.x=x.1

Hence, 1 is in C

2.

Let, u,v be in C

(u-v)x=ux-vx=xu-xv=x(u-v)

Hence, u-v is in C

3.

(uv)x=u(vx)=u(xv)=(ux)v=(xu)v=x(uv)

Hence, uv is in C

Hence, C is a subring of R

b)

Yes.

Let, x , y be in C

Since x is in C and y is in R so

xy=yx

Hence C is commutative.

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