If R is a commutative ring with unity and A is a proper ideal of R, show that R/A is a commutative ring with unity.

banganX 2020-12-17 Answered
If R is a commutative ring with unity and A is a proper ideal of R, show that RA is a commutative ring with unity.
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Expert Answer

estenutC
Answered 2020-12-18 Author has 81 answers

given R is a commutative ring with unity
ab=baa,bR
let a+A,b+ARA
(a+A)(b+A)=ab+A=ba+A=(b+A)(a+A)
RA is commutative ring (1)
Now R has unity element rArr there exists 1R so,
a1=1a=aaR
let a+ARA1R we have 1+ARA:
we have prove now 1+A is the unity element
(a+A)(1+A)=a1+A=a+A
and (1+A)(a+A)=1a+A=a+Aa+ARA:
1+A is the unity element in RA (2)
from (1) and (2) RA is a commutative ring with unity , hence proved.

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