{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T10:57:43Z","timestamp":1772794663364,"version":"3.50.1"},"reference-count":32,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2012,1,1]],"date-time":"2012-01-01T00:00:00Z","timestamp":1325376000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100008530","name":"European Regional Development Fund","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100008530","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Portuguese Government"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2012]]},"abstract":"<jats:p>The minimum depth<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>d<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>B<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>A<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>of a subring<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>B<\/mml:mi><mml:mo>\u2286<\/mml:mo><mml:mi>A<\/mml:mi><\/mml:math>introduced in the work of Boltje, Danz and K\u00fclshammer (2011) is studied and compared with the tower depth of a Frobenius extension. We show that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mi>d<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>B<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>A<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>&lt; \u221e if<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>A<\/mml:mi><\/mml:mrow><\/mml:math>is a finite-dimensional algebra and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:msup><mml:mi>B<\/mml:mi><mml:mi>e<\/mml:mi><\/mml:msup><\/mml:mrow><\/mml:math>has finite representation type. Some conditions in terms of depth and QF property are given that ensure that the modular function of a Hopf algebra restricts to the modular function of a Hopf subalgebra. If<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mi>A<\/mml:mi><mml:mo>\u2287<\/mml:mo><mml:mi>B<\/mml:mi><\/mml:math>is a QF extension, minimum left and right even subring depths are shown to coincide. If<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>A<\/mml:mi><mml:mo>\u2287<\/mml:mo><mml:mi>B<\/mml:mi><\/mml:math>is a Frobenius extension with surjective Frobenius, homomorphism, its subring depth is shown to coincide with its tower depth. Formulas for the ring, module, Frobenius and Temperley-Lieb structures are noted for the tower over a Frobenius extension in its realization as tensor powers. A depth 3 QF extension is embedded in a depth 2 QF extension; in turn certain depth<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math>extensions embed in depth 3 extensions if they are Frobenius extensions or other special ring extensions with ring structures on their relative Hochschild bar resolution groups.<\/jats:p>","DOI":"10.1155\/2012\/254791","type":"journal-article","created":{"date-parts":[[2012,10,4]],"date-time":"2012-10-04T21:05:40Z","timestamp":1349384740000},"page":"1-22","source":"Crossref","is-referenced-by-count":6,"title":["Subring Depth, Frobenius Extensions, and Towers"],"prefix":"10.1155","volume":"2012","author":[{"given":"Lars","family":"Kadison","sequence":"first","affiliation":[{"name":"Departamento de Matematica, Faculdade de Ci\u00eancias, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, 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