> [!NOTE] Theorem (Ring Product With Zero is Zero) > Let $(R,+,\times)$ be a [[Rings|ring]]. Then $\forall a\in R: 0_{R}\times a = 0_{R}= a \times 0_{R}$that is the [[Ring Zero Element|additive identity]] is a [[Ring Zero Element|zero element]] for [[Ring Product|ring product]]. *Proof*. Let $a\in R.$ Then $0\times a=(0+0)\times a=0\times a+0\times a$Using (A4), adding the additive inverse of $(0\times a),$ to both sides gives $0=0\times a-0\times a=0\times a + 0\times a-(0\times a)=0\times a.$Similarly, $a\times 0=0.$ # Applications **Theorems:** [[Product With Ring Negative]]; [[Zero Ring iff Unity Equals Zero]].