{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/68178"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/68178","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Differential Polynomial Rings: Order Properties and Morita Equivalence","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Mathis, Darrell Lee"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-14T13:09:49Z","date_published":"2014-12-14T13:09:49Z","updated_at":"2026-07-22T22:25:58Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8108599"],"render_values":[{"text":"(UMI)AAI8108599","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/68178","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Mathis, Darrell Lee"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-14T13:09:49Z","10000-01-01","1980"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/68178","(UMI)AAI8108599"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Made available in DSpace on 2014-12-14T13:09:49Z (GMT). No. of bitstreams: 1 8108599.pdf: 3418527 bytes, checksum: 93122fb7af134474b4cc808fc5d662f3 (MD5) Previous issue date: 1980","Embargo set by: Seth Robbins for item 68356 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","125 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1980."]},{"key":"dc:title","label":"Title","values":["Differential Polynomial Rings: Order Properties and Morita Equivalence"]}]}],"canonical_facts":{"dc:creator":["Mathis, Darrell Lee"],"dc:date":["2014-12-14T13:09:49Z","10000-01-01","1980"],"dc:description":["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.","Made available in DSpace on 2014-12-14T13:09:49Z (GMT). No. of bitstreams: 1 8108599.pdf: 3418527 bytes, checksum: 93122fb7af134474b4cc808fc5d662f3 (MD5) Previous issue date: 1980","Embargo set by: Seth Robbins for item 68356 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","125 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1980."],"dc:identifier":["http://hdl.handle.net/2142/68178","(UMI)AAI8108599"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Differential Polynomial Rings: Order Properties and Morita Equivalence"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:58Z"}