Let R a b c d e f with addition and multiplication defined

Let R = {a, b, c, d, e, f} with addition and multiplication defined by the following tables. One can show that R is a ring. (Do NOT try to show this.) Which element in the ring is 0_R? Is there an identity in the ring? If so, which element is 1_R^? Is R commutative? Is R an integral domain? Which elements in R are units?

Solution

a is the 0R since from the fist row and first column of the add table

a+x=x for any x in R.

2) b is the 1R see the fourth row and fourth column of multiplication table

b.x=x for any x in R.

(c) check a.c=c.a=a

a .d=d.a=a

a.e=e.a=a

a.f=f.a=a

c.d=d.c=a

c.e=e.c=c

c.f=f.c=b

d.e=e.d=b

d.f=f.d=d

Hence every element s of R is commutative that is x.y=y.x for every x,y in R.

R is commutative.

(d) integral domain if a.b=0 then either a=0 or b=0

In second table ,there is d and c non zero in R such that d.c=a which is 0R

d.c=0 but d and c are nonzero hence R is not integral domain.

(e) unit in the ring R are the elements of R which are invertible.

Only f ,b are unit

Since f.f =b ----> f is self inverse

And b is the 1R.