Another way to define a subring

Question

I was reading my class notes about subrings which said the following:

Let $A$ be a ring. We say that $B \subseteq A$ is a subring of $A$ if, with the restriction of the operations, $B$ has the structure of a ring, that is, if:

1 - it is an additive subgroup: $x, y \in B \Rightarrow x - y \in B$

2 - it is closed under multiplication: $x, y \in B \Rightarrow xy \in B$

Somehow I missread the first condition to $x, y \in B \Rightarrow x + y \in B$, which got me to question myself if with the addition sign instead of the substraction sign was also true, or in other words if the first condition was equivalent to be closed under addition.

I tried to prove it but unfourtunatetly I couldn't even prove that the additive inverse of an element in $B$ was contained in $B$.

Oddly enough I recall seeing something like this where the condition of closed under addition and mutiplication was enough to see that $B$ was a ring, with the difference that $A$ was a field instead of only being a ring.

So my questions are is this true if $A$ is a ring? And if $A$ is a field?

Without the subtraction, you need another condition to get additive inverses. — Randall, Jul 01 '24 at 21:42
Anyway, let $x=0$ to see that $-y \in B$ when $y \in B$. You also need conditions that say something like $B$ is not empty. — Randall, Jul 01 '24 at 21:43
It's common practice to prove subgroup using closure under $ab^{-1} \in H$. But any subring of a field need not be a field. Take $\Bbb{Q} \supset \Bbb{Z}$. — Daniel Donnelly, Jul 01 '24 at 21:49
See $\Bbb N$ is an additively and multiplicatively closed subset of $\Bbb R$, but not a subring. — Sassatelli Giulio, Jul 01 '24 at 21:58

Another way to define a subring

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