Question

Let $F=\left\{0,e,a,b\right\}$with operations given by the following tables. Assume associativity and distributivity and show that is a field.

Short answer

It is proved that $F$is a field.

Step-by-step solution

:Write the definition of Rings 

A ring can be defined as a non-empty set $\mathrm{ℝ}$whose elements deal with basically two operations: addition and multiplication, where elements belong to the real number $\mathrm{ℝ}$.

Property of Rings:

The given table is:

Clearly, 0 here is an additive identity andis the multiplicative identity. Also, in the table, each element satisfies the closure property as well as the property of additive inverse; thus, it is commutative.

Therefore, $F$is a ring with identity $e$.

Also, from the table, we have:

$ab=ba=e$

Thus, the product is $e$.

Hence, the given ring is commutative.

Again, we see that:

$ab=e\ne 0$

Therefore, as there are non-zero divisors, the given ring is a field.

Hence it is proved that $F$is a field.