Analyzing Complex Number Ring Structure

[sarah77]

Homework Statement

Determine whether the indicated operations of addition and multiplication are defined (closed) on the set, and give a ring structure. If a ring is formed, state whether the ring is commutative, whether it has unity, and whether it is a field: The set of all pure imaginary complex numbers ri for r $$\in$$ R with the usual addition and multiplication.

How do I begin? Please help!

 

[sarah77]

I know there are several properties that must be met in order for the set to be a ring: associative under addition and multiplication; commutative under addition; and distributive. How do I begin checking these properties the set of all pure imaginary complex numbers?

 

[I like Serena]

There's a list of 10 conditions that must hold in order for a set S with addition and multiplication to be a ring.
Do you have this list?

The first is closure under addition.
That is:

For all a, b in S, the result of the operation a + b is also in S.

So let's take 2 elements from S.
Let's say r.i and s.i, where r and s are elements of R (the real numbers), and where i is the imaginary constant.

We know that r.i + s.i = (r + s).i
So since (r + s) is an element R, this implies that (r + s).i is an element of S, which proves closure under addition.

Again, does this make sense?