1. Quick questions on Rings

R is a ring. So we have at our disposal just the axioms which make a set qualify as ring.

Q1. Is it true that a.b = 0 => b.a = 0? Personally I don't think so - but couldn't think of a quick counter example.

Q2. Assuming R has unit element, 1. Is it true that a.b = 1 => b.a = 1? My hunch again is that not necessarily.

Thanks

2. Originally Posted by aman_cc
R is a ring. So we have at our disposal just the axioms which make a set qualify as ring.

Q1. Is it true that a.b = 0 => b.a = 0? Personally I don't think so - but couldn't think of a quick counter example.
[/tex]
take $R$ to be the ring of $2 \times 2$ matrices with entries from $\mathbb{R}.$ let $a=\begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix}$ and $b=\begin{pmatrix}0 & 0 \\ 1 & 0 \end{pmatrix}.$

Q2. Assuming R has unit element, 1. Is it true that a.b = 1 => b.a = 1? My hunch again is that not necessarily.

Thanks
let $V$ be an infinite dimensional vector space over some field $F$ and $\{v_1,v_2, \cdots \}$ be a basis for $V.$ let $R=\text{End}_F(V),$ the ring of linear transformations of $V.$

define the linear maps $a,b \in R$ by: $a(v_1) = 0, \ a(v_j)=v_{j-1}, \ j \geq 2,$ and $b(v_j)=v_{j+1}, \ j \geq 1.$ then $ab = 1_R$ but $ba \neq 1_R.$ (note that $1_R$ is the identity map.)