Is the endomorphism ring of a finitely generated module over an affine Noetherian PI-ring a Jacobson ring, and must the center of such a ring be Jacobson?

Determine, for an affine Noetherian PI $k$-algebra $R$ over a field $k$, whether the endomorphism ring of every finitely generated right $R$-module must be a Jacobson ring, and whether the center $Z(R)$ must be a Jacobson ring.

Background

An associative ring is a PI-ring (polynomial identity ring) if it satisfies a polynomial identity: a nonzero polynomial $f(x_1,\dots,x_n)$ in noncommuting variables, with coefficients from a central subring, that vanishes under every substitution of ring elements; commutative rings and matrix rings over commutative rings are the basic examples. An algebra over a commutative ring $C$ is affine if it is finitely generated as a $C$-algebra, and noetherian if its left and right ideals satisfy the ascending chain condition. The Jacobson radical $J(S)$ of a ring $S$ is the intersection of its maximal left ideals, a two-sided ideal; an ideal $P$ is prime if whenever two ideals $A,B$ satisfy $AB\subseteq P$ then $A\subseteq P$ or $B\subseteq P$. A ring is a Jacobson ring if $J(S/P)=0$ for every prime ideal $P$, equivalently if every prime ideal is an intersection of primitive ideals (kernels of irreducible left modules). The noncommutative Nullstellensatz of Amitsur and Procesi states that every affine PI algebra over a field is a Jacobson ring, the direct analogue of Hilbert's Nullstellensatz for finitely generated commutative algebras.

For a right module $M$ over a ring $R$, the endomorphism ring $\mathrm{End}_R(M)$ consists of all $R$-linear maps $M\to M$. For a cyclic module (a module generated by a single element) $M=R/I$ with $I$ a right ideal, endomorphisms are exactly the left multiplications by elements of the idealizer $\mathbb{I}(I)=\{r\in R:\ rI\subseteq I\}$, the largest subring of $R$ in which $I$ is a two-sided ideal, and $\mathrm{End}_R(R/I)\cong\mathbb{I}(I)/I$; so questions about endomorphism rings of cyclic modules are questions about idealizer quotients. Classical results relate nilpotency to endomorphism rings: Levitzki proved in 1931 that nil subrings (rings in which every element is nilpotent) of the endomorphism ring of a finite-dimensional module are nilpotent (a ring or additive subset $S$ is nilpotent if $S^n=0$ for some $n$, that is, every product of $n$ elements of $S$ is zero), and Hunter showed in 1967 that nilpotence of nil subrings implies more general nilpotence, extending such conclusions to wider classes of nil subsets; Braun proved in 1984 that the Jacobson radical of a finitely generated PI ring is nilpotent. Guralnick, Small and Zelmanov (2010) completed this line: if $C$ is commutative noetherian, $R$ an affine PI-algebra over $C$, and $M$ a finitely generated right $R$-module, then every nil subring of $\mathrm{End}_R(M)$ is nilpotent, with corollaries for Lie subrings (additive subgroups closed under the commutator bracket $[a,b]=ab-ba$): a nil subset closed under the bracket generates a nilpotent subring, and a weakly Engel Lie subring of an affine PI-algebra, meaning one in which for each element $x$ some iterated commutator $[x,[x,\dots,[x,y]\dots]]$ vanishes for all $y$, is a nilpotent Lie ring (its lower central series of iterated commutators terminates).

Nilpotency of nil subrings does not imply the Jacobson property, and the Jacobson property itself fails in this generality: Wadsworth (1976) constructed a subalgebra $W$ of a finitely generated commutative algebra over a field that is not a Jacobson ring, and Resco and Wald (2007) used it to build, by an explicit triangular matrix ring idealizer construction, an affine prime PI algebra $R$ with a cyclic module whose endomorphism ring is (a quotient of) $W$ and hence not Jacobson; the same paper gives an affine prime noetherian PI ring with a cyclic module whose endomorphism ring is not noetherian. Since that ring $R$ is affine PI over the noetherian base field with a finitely generated module, no theorem under the Guralnick-Small-Zelmanov hypotheses can force $\mathrm{End}_R(M)$ to be Jacobson. What survives is the Noetherian boundary, where Resco and Wald posed the two questions: whether endomorphism rings of cyclic modules over affine noetherian PI rings are necessarily Jacobson, and whether centers of affine noetherian PI rings are necessarily Jacobson (the center $Z(R)$ is itself an endomorphism ring, since $Z(R)\cong\mathrm{End}_{R\otimes_k R^{\mathrm{op}}}(R)$ for the enveloping algebra $R\otimes_k R^{\mathrm{op}}$ of $R$, with $R^{\mathrm{op}}$ the opposite ring carrying the same elements with reversed multiplication, this being the natural ring acting on the bimodule $R$).

Problem Statement

Let $k$ be a field and let $R$ be an affine PI $k$-algebra that is noetherian; by the Amitsur-Procesi theorem $R$ is automatically a Jacobson ring. Determine whether the Jacobson property passes to the endomorphism rings of finitely generated modules, and in particular:

(1) Must $\mathrm{End}_R(M)$ be a Jacobson ring for every finitely generated right $R$-module $M$, and in particular for every cyclic module (a module generated by a single element) $M=R/I$ with $I$ a right ideal, for which $\mathrm{End}_R(R/I)\cong\mathbb{I}(I)/I$ with $\mathbb{I}(I)=\{r\in R:\ rI\subseteq I\}$ the idealizer?

(2) Companion center form: must the center $Z(R)$ of an affine noetherian PI $k$-algebra be a Jacobson ring? Since $Z(R)\cong\mathrm{End}_{R\otimes_k R^{\mathrm{op}}}(R)$ identifies the center with an endomorphism ring of $R$ as a module over its enveloping algebra, a negative answer to (2) yields a negative answer to (1), as observed by Resco and Wald.

Context and boundary: the general form of the question, with the hypotheses of Guralnick-Small-Zelmanov (2010) ($R$ an affine PI-algebra over a commutative noetherian base ring, $M$ finitely generated), is settled negatively by the Resco-Wald (2007) counterexample of an affine prime PI ring with a cyclic module whose endomorphism ring is not Jacobson; the two questions above are the surviving open core at the noetherian boundary. A natural extension, not required for closure, replaces the base field by an arbitrary noetherian Jacobson commutative base ring.

References

1. Robert M. Guralnick, Lance W. Small, Efim Zelmanov, Nil subrings of endomorphism rings of finitely generated modules over affine PI-rings, Journal of Algebra 324 (2010), no. 11, 3044-3047. DOI 10.1016/j.jalgebra.2010.06.005, https://doi.org/10.1016/j.jalgebra.2010.06.005 (zbMATH Zbl 1218.16014)
2. Richard Resco, Jeanne Wald, Endomorphism Rings, Communications in Algebra 35 (2007), no. 10, 3013-3017. DOI 10.1080/00927870701404747, https://doi.org/10.1080/00927870701404747 (zbMATH Zbl 1129.16023)
3. Adrian R. Wadsworth, Hilbert subalgebras of finitely generated algebras, Journal of Algebra 43 (1976), no. 1, 298-304. DOI 10.1016/0021-8693(76)90161-7, https://doi.org/10.1016/0021-8693(76)90161-7
4. S. A. Amitsur, C. Procesi, Jacobson-rings and Hilbert algebras with polynomial identities, Annali di Matematica Pura ed Applicata 71 (1966), 61-72. DOI 10.1007/BF02413733, https://doi.org/10.1007/BF02413733
5. Claudio Procesi, Non-commutative affine rings, Atti della Accademia Nazionale dei Lincei, Memorie, Classe di Scienze Fisiche, Matematiche e Naturali, Serie VIII, Sezione I, vol. 8 (1967), 239-255. zbMATH Zbl 0204.04802, https://zbmath.org/3323928
6. Jakob Levitzki, Über nilpotente Unterringe, Mathematische Annalen 105 (1931), 620-627. DOI 10.1007/BF01455832, https://doi.org/10.1007/BF01455832
7. Kenneth Hunter, Nilpotence of nil subrings implies more general nilpotence, Archiv der Mathematik 18 (1967), no. 2, 136-139. DOI 10.1007/BF01899638, https://doi.org/10.1007/BF01899638
8. Amiram Braun, The nilpotency of the radical in a finitely generated P.I. ring, Journal of Algebra 89 (1984), no. 2, 375-396. DOI 10.1016/0021-8693(84)90224-2, https://doi.org/10.1016/0021-8693(84)90224-2
9. Aner Shalev, On associative algebras satisfying the Engel condition, Israel Journal of Mathematics 67 (1989), no. 3, 287-290. DOI 10.1007/BF02764947, https://doi.org/10.1007/BF02764947
10. D. M. Riley, Mark C. Wilson, Associative rings satisfying the Engel condition, Proceedings of the American Mathematical Society 127 (1999), no. 4, 973-976. DOI 10.1090/S0002-9939-99-04643-2, https://doi.org/10.1090/S0002-9939-99-04643-2
11. J. C. McConnell, The Nullstellensatz and extensions of affine PI rings, Journal of the London Mathematical Society (2) 29 (1984), no. 2, 254-256. DOI 10.1112/jlms/s2-29.2.254, https://doi.org/10.1112/jlms/s2-29.2.254
12. Mátyás Domokos, Polynomial bound for the nilpotency index of finitely generated nil algebras, Algebra and Number Theory 12 (2018), no. 5, 1233-1242. DOI 10.2140/ant.2018.12.1233, https://doi.org/10.2140/ant.2018.12.1233 (arXiv:1705.01039, https://arxiv.org/abs/1705.01039)
13. J. Cimprič, M. Schötz, Left Jacobson Rings, arXiv:2503.16005 (2025), https://arxiv.org/abs/2503.16005