The set of integers Z under addition forms a group because it satisfies closure, associativity, has an identity element (0), and each element has an inverse (-n for any n).

The group of non-zero rational numbers under multiplication is abelian because for any two elements a, b ∈ Q*, ab = ba, thus satisfying commutativity.

A ring is a set equipped with two binary operations: addition and multiplication, satisfying ring axioms. An example of a commutative ring with unity is the set of integers Z.

Assume there are two identity elements, e and e'. By definition of identity, e * e' = e and e * e' = e'. Hence, e = e', proving the identity is unique.

Suppose a has two inverses, b and c. Then, b = b * e = b * (a * c) = (b * a) * c = e * c = c. Therefore, the inverse of a is unique.

The set of even integers is closed under addition, contains the identity element 0, and for every even integer, its inverse (negative) is also an even integer. Therefore, it forms a subgroup.

Yes, the set of 2x2 matrices with determinant 1, called SL(2, R), forms a group under matrix multiplication because it satisfies closure, associativity, has an identity element, and each element has an inverse.

R^n forms a module over R because it satisfies the axioms of a module: closure under addition, scalar multiplication, and distributive properties hold for R^n.

A group of prime order p has no proper subgroups and every element generates the entire group. Hence, the group is cyclic.

The set of complex numbers under addition forms a group because it satisfies closure, associativity, has an identity element (0), and every element has an inverse (-z for any z).