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Math Help - Quick questions on Rings

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    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.

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    Quote Originally Posted by aman_cc View Post
    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.)
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