{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71232"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71232","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Generalized Clifford Theory (Group Ring, Frobenius Extensions)","abstract":"Let R and S be two rings with identity elements and let l : R (--->) S be a ring homomorphism preserving their identity elements. Then any S-module W can be regarded as an R-module W(,R). Also, for any right R-module V, we can form the "induced" S-Module V('S) = V (CRTIMES)(,R) S. Fixing a right R-module V, we let A = End(,R)(V) and B = End(,S)(V('S)). Then there is a natural ring homomorphism l' : A (--->) B. The R-module V is said to be weakly S-invariant if V('S) is isomorphic to a direct summand of a direct sum of a finite number of copies of V as R-modules. (We say that (V('S))(,R) weakly divides V in this case.)","abstract_html":"Let R and S be two rings with identity elements and let l : R (--->) S be a ring homomorphism preserving their identity elements. Then any S-module W can be regarded as an R-module W(,R). Also, for any right R-module V, we can form the "induced" S-Module V('S) = V (CRTIMES)(,R) S. Fixing a right R-module V, we let A = End(,R)(V) and B = End(,S)(V('S)). Then there is a natural ring homomorphism l' : A (--->) B. The R-module V is said to be weakly S-invariant if V('S) is isomorphic to a direct summand of a direct sum of a finite number of copies of V as R-modules. (We say that (V('S))(,R) weakly divides V in this case.)","abstract_has_math":false,"creators":["Uno, Katsuhiro"],"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":"2014-12-16T06:18:12Z","date_published":"2014-12-16T06:18:12Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8511681"],"render_values":[{"text":"(UMI)AAI8511681","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71232","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Uno, Katsuhiro"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:12Z","10000-01-01","1985"]},{"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":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71232","(UMI)AAI8511681"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let R and S be two rings with identity elements and let l : R (--->) S be a ring homomorphism preserving their identity elements. Then any S-module W can be regarded as an R-module W(,R). Also, for any right R-module V, we can form the "induced" S-Module V('S) = V (CRTIMES)(,R) S. Fixing a right R-module V, we let A = End(,R)(V) and B = End(,S)(V('S)). Then there is a natural ring homomorphism l' : A (--->) B. The R-module V is said to be weakly S-invariant if V('S) is isomorphic to a direct summand of a direct sum of a finite number of copies of V as R-modules. (We say that (V('S))(,R) weakly divides V in this case.)","The notion of weak invariance is introduced in Chapter 2. In Chapter 3, we prove a correspondence theorem, which generalizes the classical Clifford theorem studied by several authors such as Clifford, Cline, Conlon, Dade and Tucker. Let Mod(S(VBAR)V) denote the category whose objects are all right S-modules W such that W(,R) weakly divides V and whose morphisms are all S-homomorphisms among those modules. Likewise, we define Mod(B(VBAR)A). Then the theorem says that the two additive functors (.) (CRTIMES)(,R) S and Hom(,S)(V('S),(.)) form an equivalence between Mod(S(VBAR)V) and Mod(B(VBAR)A).","The map l : R (--->) S is called Frobenius if S(,R) is finitely generated projective and S (TURNEQ) Hom(,R)(S(,R),R(,R)) as (R,S)-bimodules. Such ring homomorphisms are studied in Chapter 4. They satisfy the "other" Frobenius Reciprocity Law. Also, it is shown that if l : R (--->) S is Frobenius and V is weakly S-invariant, then l' : A (--->) B is Frobenius.","Let l(,1) : S (--->) T be another homomorphism of rings. Let C = End(,T)(V('T)) with A and B being the same as before. Suppose that V is weakly S- and T-invariant and that V('S) is weakly T-invariant. Assuming that l(,1) : S (--->) T is Frobenius, we prove that an object W of Mod(S(VBAR)V) is weakly T-invariant if and only if the object Y of Mod(B(VBAR)A) corresponding to W is weakly C-invariant. We then establish the compounding of Clifford correspondences constructed in Chapter 3. This is done in Chapter 5.","Made available in DSpace on 2014-12-16T06:18:12Z (GMT). No. of bitstreams: 1 8511681.pdf: 1826869 bytes, checksum: a968d3ef0ce19d790d691a8da6c6ebe8 (MD5) Previous issue date: 1985","Embargo set by: Seth Robbins for item 71398 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","70 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985."]},{"key":"dc:title","label":"Title","values":["Generalized Clifford Theory (Group Ring, Frobenius Extensions)"]}]}],"canonical_facts":{"dc:creator":["Uno, Katsuhiro"],"dc:date":["2014-12-16T06:18:12Z","10000-01-01","1985"],"dc:description":["Let R and S be two rings with identity elements and let l : R (--->) S be a ring homomorphism preserving their identity elements. Then any S-module W can be regarded as an R-module W(,R). Also, for any right R-module V, we can form the "induced" S-Module V('S) = V (CRTIMES)(,R) S. Fixing a right R-module V, we let A = End(,R)(V) and B = End(,S)(V('S)). Then there is a natural ring homomorphism l' : A (--->) B. The R-module V is said to be weakly S-invariant if V('S) is isomorphic to a direct summand of a direct sum of a finite number of copies of V as R-modules. (We say that (V('S))(,R) weakly divides V in this case.)","The notion of weak invariance is introduced in Chapter 2. In Chapter 3, we prove a correspondence theorem, which generalizes the classical Clifford theorem studied by several authors such as Clifford, Cline, Conlon, Dade and Tucker. Let Mod(S(VBAR)V) denote the category whose objects are all right S-modules W such that W(,R) weakly divides V and whose morphisms are all S-homomorphisms among those modules. Likewise, we define Mod(B(VBAR)A). Then the theorem says that the two additive functors (.) (CRTIMES)(,R) S and Hom(,S)(V('S),(.)) form an equivalence between Mod(S(VBAR)V) and Mod(B(VBAR)A).","The map l : R (--->) S is called Frobenius if S(,R) is finitely generated projective and S (TURNEQ) Hom(,R)(S(,R),R(,R)) as (R,S)-bimodules. Such ring homomorphisms are studied in Chapter 4. They satisfy the "other" Frobenius Reciprocity Law. Also, it is shown that if l : R (--->) S is Frobenius and V is weakly S-invariant, then l' : A (--->) B is Frobenius.","Let l(,1) : S (--->) T be another homomorphism of rings. Let C = End(,T)(V('T)) with A and B being the same as before. Suppose that V is weakly S- and T-invariant and that V('S) is weakly T-invariant. Assuming that l(,1) : S (--->) T is Frobenius, we prove that an object W of Mod(S(VBAR)V) is weakly T-invariant if and only if the object Y of Mod(B(VBAR)A) corresponding to W is weakly C-invariant. We then establish the compounding of Clifford correspondences constructed in Chapter 3. This is done in Chapter 5.","Made available in DSpace on 2014-12-16T06:18:12Z (GMT). No. of bitstreams: 1 8511681.pdf: 1826869 bytes, checksum: a968d3ef0ce19d790d691a8da6c6ebe8 (MD5) Previous issue date: 1985","Embargo set by: Seth Robbins for item 71398 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","70 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985."],"dc:identifier":["http://hdl.handle.net/2142/71232","(UMI)AAI8511681"],"dc:subject":["Mathematics"],"dc:title":["Generalized Clifford Theory (Group Ring, Frobenius Extensions)"],"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":"2026-07-22T22:26:04Z"}