noncommutative algebra - Semiprime (but not prime) ring whose center is a domain

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 $I=I_1cap I_2$ 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.

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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.