Find all units in each given ring

[sarah77]

Homework Statement

Describe all the units (if any) in each given ring: 2Z X Z with addition and multiplication by components; and Z X Q X Z with addition and multiplication by components

Homework Equations

The Attempt at a Solution

I do not know how to begin, I am not sure how to find the units in 2Z and Z

 

[micromass]

You can start by showing that (a,b) is a unit in AxB if and only if a and b are units...

 

[I like Serena]

Let's start with the definition of a unit in a ring:

In mathematics, an invertible element or a unit in a (unital) ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that
uv = vu = 1, where 1 is the multiplicative identity element.


In Z2 we have only two elements: 0 and 1.
Which of those satisfies the definition of a "unit"?

Do you know how to multiply in 2Z X Z?
If so, What is the multiplicative identity element in 2Z X Z?

 

[sarah77]

Let's start with the definition of a unit in a ring:

In mathematics, an invertible element or a unit in a (unital) ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that
uv = vu = 1, where 1 is the multiplicative identity element.


In Z2 we have only two elements: 0 and 1.
Which of those satisfies the definition of a "unit"?

Do you know how to multiply in 2Z X Z?
If so, What is the multiplicative identity element in 2Z X Z?

The Unit in 2Z is 1, so the multiplicative identity element in 2Z x Z is 1; does that mean that all numbers are units in 2Z x Z?

 

[I like Serena]

The Unit in 2Z is 1, so the multiplicative identity element in 2Z x Z is 1; does that mean that all numbers are units in 2Z x Z?

Not so fast! ;)

Since 0.v = 0 for every v in Z2, 0 is not a unit in Z2, and so yes, the unit in Z2 is 1.

However, 1 is not the multiplicative identity element in Z2 x Z.
This set is a so called "cartesian product" (may be you already know all of this, so excuse me if I'm too explicit).
Each element in Z2 x Z is of the form (a, b), where a is an element of Z2 and b is an element of Z.
And multiplication, for instance, is defined by:
(a, b) x (c, d) = (ac mod 2, bd)

In particular the multiplicative identity element in Z2 x Z is (1, 1).

A "unit" is, however, something different from the multiplicative identity element.
It is any element (a, b) in Z2 x Z, for which there exists a (c, d), such that:
(a, b) x (c, d) = (1, 1).

This in turn means that: ac mod 2 = 1 mod 2
and it also means that: bd = 1

Since the only (a, b) for which this is possible is (1, 1), this means that (1, 1) is the only "unit" in Z2 x Z.

Does this make sense to you?