Set or Ring, and group of units?

Question

I have a couple of questions. I understand the axioms needed for a ring. But am confused about a unitary ring? does this just mean its a ring but has to have the unit 1?

Also I do not understand what the set of integers * natural numbers would be?

Finally, what are its units? For questions like this would it just be the whole group? I understand a unit is an element with a multiplicative inverse and how to do it for finite integer groups just not for infinite groups like above.

Really grateful for help.

Answer 1

Yes that is what a unitary ring is, a ring with unit $1$.

An element of this set looks like $(-5,2)$, or any such ordered pair where the first entry is an integer and the second is a natural number.

We also know $1,-1$ are units of $\mathbb{Z}$ and we know $1$ is the only unit in $\mathbb{N}$. How can we use this to find the units in the product?

Answer 2

A unitary ring is a ring containing the element $1$, a neutral element with respect to multiplication. (This is not required of all rings.)

$\mathbb{Z} \times \mathbb{N}$ is the set of ordered pairs $(z,n)$ where $z$ is an integer and $n$ is a natural number.

Units are invertible elements, meaning the set of elements such that there exists a multiplicative inverse for each element: $\{(z,n) \in \mathbb{Z}\times \mathbb{N} \mid \exists (y,m) \in \mathbb{Z}\times \mathbb{N} \text{ such that } (z,n)\cdot_R(y,m) = (zy, nm) = (1,1)\}$. As a hint: is there anything in $\mathbb{Z}$ that when multiplied by $2$ gives $1$? Does this change if it's $(2,1) \in \mathbb{Z}\times \mathbb{N}$?