7
$\begingroup$

Let $R$ be a unital ring and denote its center by $Z(R)$. If $I$ is an ideal of $Z(R)$, then the set $RI$ (consisting of finite sums of elements of the form ra where $r\in R$ and $a\in I$) is clearly an ideal of $R$.

My question is the following:
If $I$ is a proper ideal of $Z(R)$, is $RI$ necessarily a proper ideal of $R$?

The proof that I had in mind does not seem to work out, and I am now suspecting that the answer is negative. Are there any nice and intuitive counter-examples?

  • 0
    Dear Per: I think Parsa's answer is incorrect. Here is my argument: Assume that $I\subset Z:=Z(R)$ is principal, generated by $z\in Z$, and that $RI=R$, that is $Rz=R$. Then $z$ is invertible in $R$, and thus also in $Z$.2012-02-19

1 Answers 1

3

I'm new to non-commutative algebra, but maybe the following counterexample works. Take the noncommutative polynomial ring $k$ over a field $k$, and mod out by the following relations: $XZ-ZX, YZ-ZY,$ and $XZ+YZ-1$. I guess by mod out by these relations I mean go modulo the two-sided ideal generated by these three elements, and denote the quotient ring $R$. Then the center is $Z(R)=k[Z]$. Take $I=(Z) \subsetneq Z(R)$. However, we have $RI=R$ because $XZ+YZ=1$.

  • 0
    Dear @Parsa: Thanks for having answered my comment. I found your comment by chance. Please use the `@` sign if you write another comment for me. - I'd be happy to know if you agree with this: The ring $R$ can be defined by $R:=\frac{k[Z]\langle X,Y\rangle}{(XZ+YZ-1)}\quad.$ Write $x,y,z$ for the images of $X,Y,Z$ in $R$. We have $(x+y)z=1$. That is, $z$ is invertible and its inverse is $x+y$. Thus, $R$ is commutative, and isomorphic to $k[X,Z,Z^{-1}]$.2012-02-18