Differential Polynomial Rings: Order Properties and Morita Equivalence

Mathis, Darrell Lee

Permalink

https://hdl.handle.net/2142/68178 Copy

Description

Title

Differential Polynomial Rings: Order Properties and Morita Equivalence

Author(s)

Mathis, Darrell Lee

Issue Date

1980

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

Abstract

• The thesis is concerned with the behavior of certain ring properties with respect to the ring extension R (--->) R{X,(delta)}, where the latter is the ring of differential polynomials. Three properties are considered.
• The first property is Morita Equivalence. Let F : Mod-R (--->) Mod-S be an equivalence of module categories. Also let D(R) denote the Lie ring of derivations on R modulo the ideal of inner derivations on R. Then F induces a Lie ring isomorphism r(, ): D(R) (--->) D(S). For each derivation (delta)
• on R, let (delta) denote the image of (delta) in D(R). Then (delta) determines a ring of(, ) differential polynomials, denoted by R{X,(delta)}, up to ring isomorphism. Let U(,R)(' ): Mod-R{X,(delta)} (--->) Mod-R be the forgetful functor induced by the canonical ring map R(' )(--->) R{X,(delta)}. If (lamda) = (OMEGA)((delta)), where (delta) is a derivation on
• R and (lamda) is a derivation on S, then F induces an equivalence F(' ): Mod-R{X,(delta)} (--->)(' )Mod-S{X,(lamda)} such that U(,S(DEGREES)) F = F (,(DEGREES)) U(,R). Moreover, the lattice isomorphism from the lattice of ideals of R to the lattice of ideals of S induces an isomorphism from the lattice of (delta)-invariant ideals of R to the(' )(lamda)-invariant ideals of S.(' )
• The second property is that of being a right order in a right Artinian ring. Using Block's characterization of (delta)-simple rings with a minimal ideal, it is shown that, if R is a right order in a right Artinian ring, then R{X,(delta)} is also. Moreover, the multiplicative set of polynominals with regular leading coefficient is an exhaustive set.
• Finally, we consider orders in quasi-Frobenius rings (QF rings). Assuming that R is of a QF ring, we show that the right Goldie dimension of R{X,(delta)} equals the length of R/N(,(delta))(R), where N(,(delta))(R) is the (delta)-prime radical of R. From this it follows that, if R is a right order in a QF ring, then R{X,(delta)} is also. Let Q(,cl)(R) denote the right quotient ring of R, if it
• exists. Other consequences are the following: (a) If Q(,cl) (R) is right Artinian, then Q(,cl) (R) and Q(,cl)(R{X,(delta)}) have the same length. We also determine the structure of Q(,cl)(R{X,(delta)}) modulo its prime radical in terms of Q(,cl)(R). (b) If R is right Noetherian and (delta)-semiprime, then Q(,cl)(R) is a QF ring. (c) If Q(,cl)(R) is a QF ring, then Q(,cl)(R/N(,(delta))(R)) is a QF ring.
• Finally, we consider some partial converses for the last two properties. If Q(,cl)(R{X,(delta)}) is right Artinian (QF), then Q(,cl)(R) is right Artinian (QF) if any of the following conditions hold: (i) The multiplicative set of polynomials with regular leading coefficient is an exhaustive set, (ii) N(R) is a (delta)-invariant ideal, (iii) R is right Noetherian, or (iv) R is commutative.

Graduation Semester

1980

Type of Resource

text

Permalink

http://hdl.handle.net/2142/68178

Owning Collections