Show that (ZZ_6 +_6, xx_6) is a commutative ring. Is (ZZ_6 +_6, xx_6) a field?

Braxton Pugh 2020-12-03 Answered
Show that (Z6+6,×6) is a commutative ring. Is (Z6+6,×6) a field?
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Answered 2020-12-04 Author has 108 answers

Since addition modulo n and multiplication modulo n of integers are commutative . So, the ring (Z6+6,×6) is a commutative ring.
Suppose a,bZ6
Then, ab=ba=m , where m is the reminder obtained when ab, ba is divided by 6 and 0m5
ab mmod6,ba mmod6
Also, a+b=b+a=n , where n is the reminder obtained when a+b,b+a is divided by 6 and 0n5
a+b nmod6,b+a nmod6
Hence, Z6 is a commutative ring
A ring is a field if its every element has an inverse.
As 2,3Z6
but 23=6mod6=0
So, 2 and 3 are zero divisors and do not have inverse .
So, (Z6+6,×6) is not a field.

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