Exercise 1. This software determines the number of

Step 1 of 3

A ring R is a set with two binary operations, addition and multiplication, satisfying several properties: R is an Abelian group under addition, and the multiplication operation satisfies the associative law

And distributive laws

And

For every

The identity of the addition operation is denoted 0. If the multiplication operation has an identity, it is called a unity. If multiplication is commutative, we say that R is commutative. Let A be a subring of a ring R. If, for every  and

And

We say that A is an ideal of R.

If A is an ideal of R, the set of cosets under addition.

Is itself a ring. Addition is defined as

This ring is called the factor ring and denoted.

The ring of Gaussian integers is defined as  

Under ordinary addition and multiplication of complex numbers. If a is an element of a commutative ring R with unity, the principal ideal generated by a is