**On the nature and number of the noncommutative minimal ring extensions of a field **

**David E. Dobbs**

Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320

Pages 272–290 | Received 06 June 2023, Accepted 05 September 2023, Published 11 December 2023

**Abstract**

It is known that if k is a field, then any noncommutative minimal ring extension of k is either a prime ring or a non-semiprime ring. Our results include the following. If R is a ring, the collection of R-isomorphism classes represented by minimal ring extensions of R is a set. If X is an indeterminate over a field k, the cardinal number of the set of k(X)-isomorphism classes represented by noncommutative minimal ring extensions of k(X) is infinite, at least denumerably many of those classes have representatives which are simple (and left- and right-Artinian) rings (and, hence, prime rings), and if one also assumes that the field k is infinite, then at least denumerably many of those classes have representatives which are non-semiprime rings. If k is any algebraically closed field (more generally, a field with infinitely many automorphisms), the set of k-isomorphism classes represented by noncommutative minimal ring extensions of k is infinite and the representatives of infinitely many of those classes are non-semiprime rings. If k is a finite field of characteristic p and of cardinality pn with n > 1, the cardinal number of the set of k-isomorphism classes represented by noncommutative minimal ring extensions of k is at least n, and at least n−1 of those classes can be represented by a non-semiprime ring. If p is a prime number, then for finite fields k of characteristic p, there is no absolute finite upper bound on the cardinal number of the set of k-isomorphism classes represented by noncommutative minimal ring extensions of k that are simple (hence, prime) rings.

**Keywords**: Unital associative ring, minimal ring extension, field, simple ring, prime ring, noncommutative, field extension, finite field, algebraically closed field, Dorroh extension, rng, bimodule, Lüroth’s Theorem, skew polynomial ring.

**MSC numbers**: Primary 16B99; Secondary 12F05, 12F10, 16S50, 16D20, 13B99, 12F20, 16P10.

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