RINGS - prove:(1)multiplicative identity is unique, (2)left & right multiplicative inverses are =

I need to understand how to show/prove the following regarding a ring R: 1) if a ring R has a multiplicative identity, then the multiplicative identity is unique. 2) if an element r that is in the ring R has a left multiplicative inverse r' and a right multiplicative inverse r", then r' = r".

Show that a set of matrices is a ring without identity element

I am given the set of infinite-by-infinite matrices with real entries that have only finitely many nonzero entries. Also, the matrix has entries a_ij, where i and j are natural numbers and there is a natural number n such that a_ij = 0 if i >= n or j >= n. I need to understand how to show that this set of matrices is a ring without identity element.

Show that a ring isomorphism has a multiplicative identity and is commutative

Suppose φ:R --> S is a ring isomorphism. Show that R has a multiplicative identity if, and only if, S has a multiplicative identity. Show that R is commutative if, and only if, S is commutative.

Show that the set of real rational functions is a field.

NOTE: In this description, R represents the symbol for the set of real numbers. I couldn't find a way to type or copy the correct R symbol for the set of real numbers. Also, the parentheses in R(x) is used to distinguish the ring R(x) of rational functions from the ring R[x] of polynomials. Show that the set R(x) of rational functions p(x)/q(x), where p(x), q(x) are in R[x] and q(x) <> 0, is a field.

Rings: left and right ideals

Show that if R is a ring with identity element and x is an element in R, then Rx = {rx: r in R} is the principal left ideal generated by x. Similarly, xR = {xr: r in R} is the principal right ideal generated by x.