Provide a solution for the following problems.

The following are properties that were developed as a result of a study conducted on all the Groups as well as rings in a field. Ensure you have a deep understanding on the entities of these unit since you will be required to answer several questions from the given properties. These are the properties to be considered;

G-i) ClosureR- i) Closure under multiplication

G- ii) AssociateR- ii) Associativity of multiplication

G- iii) Identity ElementR- iii) Distributive law

G- iv) Inverse ElementR- iv) Commutativity of multiplication

G- v) CommutativeR- v) Multiplicative Identity

R- vi) No zero divisors

R- vii) Multiplicative Inverse

Determine the following.

The properties that are satisfied by all the groups.

The properties satisfied by an Abelian group

The property represented by a.(b.c) = (a.b).c

The properties that are satisfied by an Integral Domain.

Is it true that all the properties above from G-i to R-vi are satisfied by a field?

Under which properties is a ring said to be commutative?

Determine the properties satisfied by a ring.

The property represented by a(b+c) = ac+bc