Question #294457

Let A, B be two subrings of a ring R such that for all a e A, b e B,


ba e A then show that


(1) A + B is a subring of R.


(2) A is an ideal of A + B.


3) An B is an ideal of B.

1
Expert's answer
2022-02-09T10:39:38-0500

Part 1


Suppose A and B are two subrings such that

A+BA+B is a subring if BAB\sube\>A

aA\therefore\exist\>a\in\>A such that aBa\notin\>B

bB\exist\>b\in\>B such that bAb\in\>A


A+BA+B satisfies the axioms

(i)A+B(i)A+B\ne\empty

0RA0_R\in\>A and 0RB0_R\in\>B


    A+B\implies\>A+B is not empty


(ii)(ii)

abA+Ba-b\in\>A+B

If abAa-b\in\>A then a(ab)A    bAa-(a-b)\in\>A\implies\>b\in\>A


(iii)(iii)

a,bA+B,a.bA+B\forall\>a,b\in\>A+B,a.b\in\>A+B


Part 2

A is a subring of R     A\implies\>A\ne\empty


If a,bAa,b\in\>A then abA+Ba-b\in\>A+B

If rRr\in\R and aA+Ba\in\>A+B then ar,raA+Bar,ra\in\>A+B


    A\implies\>A is an ideal of A+BA+B


Part 3

(i)(i)

A,BA,B\ne\empty

    AnB\implies\>AnB\ne\empty


(ii)a,bAnB    abAnB(ii)a,b\in\>AnB\implies\>a-b\in\>AnB


(iii)rR,bAnB    br,rbAnB(iii)r\in\>R,b\in\>AnB\implies\>br,rb\in\>AnB


Therefore AnBAnB is an ideal of BB



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