# zero times an element is zero in a ring

###### Lemma 1.

Let $R$ be a ring with zero element $0$ (i.e. $0$ is the additive identity of $R$). Then for any element $a\in R$ we have $0\cdot a=a\cdot 0=0$.

###### Proof.
 $\displaystyle 0\cdot a$ $\displaystyle=$ $\displaystyle(0+0)\cdot a,\quad\text{ by definition of zero}$ $\displaystyle=$ $\displaystyle 0\cdot a+0\cdot a,\quad\text{ by the distributive law}$

Thus $0\cdot a=0\cdot a+0\cdot a$. Let $b$ be the additive inverse of $0\cdot a\in R$. Hence:

 $\displaystyle b+0\cdot a=b+(0\cdot a+0\cdot a)$ $\displaystyle(b+0\cdot a)=(b+0\cdot a)+0\cdot a$ $\displaystyle 0=0+0\cdot a$ $\displaystyle 0=0\cdot a$

as claimed. The proof of $a\cdot 0=0$ is done analogously. ∎

Title zero times an element is zero in a ring ZeroTimesAnElementIsZeroInARing 2013-03-22 14:13:57 2013-03-22 14:13:57 alozano (2414) alozano (2414) 8 alozano (2414) Theorem msc 20-00 msc 16-00 msc 13-00 $0\cdot a=0$ 1cdotAA AbsorbingElement