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The center of a prime ring is a domain and the center of a semiprime ring is reduced.

Now I have no evidence to believe that if the center of a semiprime ring R is a domain,

then R has to be a prime ring. So I'm looking for some examples of semiprime rings R

with these properties: R is not prime and the center of R is a domain.

This is not important but it'd be nice if the center of R is not too small.

Thanks for reading

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I find it preposterous to completely answer a technical question providing an explanation and references, and get exactly ZERO response. –  Victor Protsak Aug 5 '10 at 16:42
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1 Answer

A whole class of examples of this kind can be obtained from prime ideals in enveloping algebras with the same central character. I will sketch the construction in the case of primitive ideals in simple Lie algebras, but these conditions can be considerably relaxed.

Let $\mathfrak{g}$ be a complex simple Lie algebra of rank at least $2$ (i.e. not $\mathfrak{sl}_2$), $U(\mathfrak{g})$ its universal enveloping algebra, $I_1, I_2$ two incomparable primitive ideals with the same infinitesimal character, and $I=I_1\cap I_2$ their intersection. Then $A=U(\mathfrak{g})/I$ is semiprimitive, and hence semiprime. By the assumption, $I_1$ and $I_2$ intersect $Z(\mathfrak{g})$ at the same maximial ideal, so $Z(A)=\mathbb{C},$ which is a domain. To get a larger center, you can repeat this construction with incomparable prime ideals whose intersection with $Z(\mathfrak{g})$ is the same non-maximal prime ideal of the latter ring.


If you know representation theory of simple Lie algebras, here is an explicit construction of a pair of ideals with these properties: let $\lambda$ be an integral dominant weight, choose two different simple reflections $s_i, i=1, 2$ in the Weyl group, and let $I_i=\text{Ann}\ L(s_i*\lambda)$ be the annihilator of the simple highest weight module with highest weight $s_i(\lambda+\rho)-\rho.$ The ideals $I_1$ and $I_2$ have the same infinitesimal character by the Harish-Chandra isomorphism and they are incomparable by the theory of $\tau$-invariant: $\tau(I_i)=\{s_i\}$, but $\tau$-invariant is compatible with the containment of primitive ideals.

Everything except for the $\tau$-invariant is explained in Dixmier's "Enveloping algebras", and you can find the rest in Borho–Jantzen's or Vogan's old papers (you need the main property of the $\tau$-invariant stated above) or read Jantzen's book about the enveloping algebras (in German) for the whole story.

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