Let R be the ring of all 2 x 2 matrices with integer entries. Which of the following subsets of R is an integral domain?a)b)c)d)Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Since in a integral domain property of commutativity an d o f unity m u s t be satisfied.

is unity when x = 1, it is also commutative.
It does not have zero divisor, so it will be integral domain.
In case of
a)

= 0 (zero divisor)
b)

=0 (zero divisor)
d)

= 0 (zero divisor)

Answer

In group theory, a group is a set equipped with a binary operation that satisfies four fundamental properties - closure, associativity, identity element, and inverse element. A group must have exactly one identity element. This is because for any element 'a' in the group, there must exist an inverse 'b' such that the product 'a * b' (or 'b * a') equals the identity element. If there were more than one identity element, this property would be violated, and the set would not form a group.

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