The subring criterion

Question

As we know that the subring criterion states that a subset $H$ of ring $R$ is a subring if and only if :

(1) $H$ is non-void , and

(2) for all $x,y \in H$,$x-y \in H$.

(3) product $xy \in H$ .

The thing I can't understand is why we need to ensure that product $xy \in H$?Even if we don't ensure that what will happen for $H$ to be a subring.I don't find it intutive because $R$ is not a group under multiplication. Please help...

Answer 1

We want $H$ to also be a ring, so our multiplication can't let us get "outside" it. For example, let $R = \mathbb{Z}[x]$ and $H$ be all elements of the form $ax + b$ (for $a, b \in \mathbb{Z})$. This is a perfectly valid sub group of $R$, but it's not a ring: $x \in H$, but $x \cdot x$ is not.

If you know what monoids are, another way to think of the subring criterion is this: $(H, +)$ must be a subgroup of $(R, +)$, and $(H, \cdot)$ must be a submonoid of $(R, \cdot)$. (If you don't require your rings to have an identity, replace "monoid" with "semigroup").

Answer 2

A ring needs to have two laws $+$ and $\times$. The condition $(3)$ allow us the second law for $H$.