Add description for 169 "Kerr's goldie ring"

[rschwiebert]

https://ringtheory.herokuapp.com/rings/ring/169/

J. W. Kerr. An example of a Goldie ring whose matrix ring is not Goldie. (1979)

[dyunov]

Do we want $M_2(R)$ as a separate entry or not?

Description

Let $K$ be a commutative integral domain (we take $K = \Bbb Q$). Define two countably sets of variables $A = {a_i, b_i, c_i, d_i}$ and $X = {x_i, y_i, u}$, where $i$ runs through positive integers. Consider the graded $K$-algebras $T = K[A]$, $D = K[X]$ where the indexed variables have degree $1$ and $u$ has degree $0$.
The kernel $P$ of the homomorphism $f$: $T \to D$, $a_i \mapsto ux_i$, $b_i \mapsto x_i$, $c_i \mapsto y_i$, $d_i \mapsto uy_i$, is a homogeneous prime ideal of $T$; let $P_i$ be its homogeneous component of degree $i \ge 0$. The component $P_2$ is a free $K$-module with basis $B = {\gamma_{ij} = a_i c_j - b_i d_j, a_i b_j - a_j b_i, c_i d_j - c_j d_i}$ for all $i,j$.
Define the ideal $I \lhd T$ generated by ${ P_i}$ for $i \ge 3$, $B \setminus { \gamma_{ii} }{i=1}^\infty$, and ${\gamma{ii} - \gamma_{jj}}$ for all $i,j$. The ring is $R = T/I$.

I dropped the overline from the ring name and changed Kerr's $R$ to $T$.

Properties❌✔

• ❌ semiprime. Use the description of nontrivial annihilator ideals: the only ones are $\overline{P} \subset \overline{Q}$ with $Ann(\overline{P}) = \overline{Q}$. (Or see the very first paragraph of Kerr's paper.)
• ✔ nonzero socle. As $\overline{P}$ is isomorphic to $K$ as a $K$-module, it's simple as an $R$-module too.
• ✔ connected. We have $R = \Bbb Q + \overline{Q}$ (unfortunate notation). If something is idempotent, so is its degree 0 component. In a graded ring, $f_0 + f_i + \ldots + f_n$ (homogeneous components) could only be idempotent if $f_0^2 = f_0$ and $f_i = 2 f_i f_0$, i.e. $f_i(1 - 2f_0) = 0$. But our degree 0 component is a field, and other components are unital modules over it, so either $1 = 2f_0$ (contradiction with $f_0^2 = f_0$) or $f_i = 0$.
• (added) ❌ local. $R = T/I$, $I \subset P$; $T/P \subset K[X]$ is a domain that is not a field.

Comments

• Surely the Jacobson radical contains $\overline{P}$. I don't know if there is an equality.
• The last paragraph of the paper might help with Armendariz but I'm too lazy for that.

[rschwiebert]

Do we want $M_2(R)$ as a separate entry or not?

Yes, that is my intention now that relations are a thing :)