Homological Properties of (Graded) Noetherian PI Rings
Let R be a connected, graded, Noetherian PI ring. If injdim( R) = n < ∞, then we prove that R is Auslander-Gorenstein and Cohen-Macaulay, with Gelfand-Kirillov dimension equal to n. If gldim( R) = n < ∞, then R is a domain, finitely generated as a module over its centre and a maximal order in...
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| Veröffentlicht in: | Journal of algebra 1994-09, Vol.168 (3), p.988-1026 |
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| container_title | Journal of algebra |
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| creator | Stafford, J.T. Zhang, J.J. |
| description | Let
R be a connected, graded, Noetherian PI ring. If injdim(
R) =
n < ∞, then we prove that
R is Auslander-Gorenstein and Cohen-Macaulay, with Gelfand-Kirillov dimension equal to
n. If gldim(
R) =
n < ∞, then
R is a domain, finitely generated as a module over its centre and a maximal order in its quotient division ring. Similar results hold if
R is assumed to be local rather than connected graded. Alternatively, suppose that
R is a Noetherian PI ring with gldim(
R) < ∞ such that hd(
R/
M
1) = hd(
R/
M
2) for any two maximal ideals
M
i
in the same clique. Then,
R is a direct sum of prime rings, is integral over its centre, and is Auslander-Gorenstein. If
R is a prime ring, then the centre
Z(
R) of
R is a Krull domain and
R equals its trace ring TR. Moreover, hd(
R/
M) = height(
M), for every maximal ideal
M of
R. |
| doi_str_mv | 10.1006/jabr.1994.1267 |
| format | Article |
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R be a connected, graded, Noetherian PI ring. If injdim(
R) =
n < ∞, then we prove that
R is Auslander-Gorenstein and Cohen-Macaulay, with Gelfand-Kirillov dimension equal to
n. If gldim(
R) =
n < ∞, then
R is a domain, finitely generated as a module over its centre and a maximal order in its quotient division ring. Similar results hold if
R is assumed to be local rather than connected graded. Alternatively, suppose that
R is a Noetherian PI ring with gldim(
R) < ∞ such that hd(
R/
M
1) = hd(
R/
M
2) for any two maximal ideals
M
i
in the same clique. Then,
R is a direct sum of prime rings, is integral over its centre, and is Auslander-Gorenstein. If
R is a prime ring, then the centre
Z(
R) of
R is a Krull domain and
R equals its trace ring TR. Moreover, hd(
R/
M) = height(
M), for every maximal ideal
M of
R.</description><identifier>ISSN: 0021-8693</identifier><identifier>EISSN: 1090-266X</identifier><identifier>DOI: 10.1006/jabr.1994.1267</identifier><language>eng</language><publisher>Elsevier Inc</publisher><ispartof>Journal of algebra, 1994-09, Vol.168 (3), p.988-1026</ispartof><rights>1994 Academic Press</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c326t-6ff858d03b1504654fcf454e4b351001c6c16d9975cf9e4a5620dde8821250b83</citedby></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1006/jabr.1994.1267$$EHTML$$P50$$Gelsevier$$Hfree_for_read</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids></links><search><creatorcontrib>Stafford, J.T.</creatorcontrib><creatorcontrib>Zhang, J.J.</creatorcontrib><title>Homological Properties of (Graded) Noetherian PI Rings</title><title>Journal of algebra</title><description>Let
R be a connected, graded, Noetherian PI ring. If injdim(
R) =
n < ∞, then we prove that
R is Auslander-Gorenstein and Cohen-Macaulay, with Gelfand-Kirillov dimension equal to
n. If gldim(
R) =
n < ∞, then
R is a domain, finitely generated as a module over its centre and a maximal order in its quotient division ring. Similar results hold if
R is assumed to be local rather than connected graded. Alternatively, suppose that
R is a Noetherian PI ring with gldim(
R) < ∞ such that hd(
R/
M
1) = hd(
R/
M
2) for any two maximal ideals
M
i
in the same clique. Then,
R is a direct sum of prime rings, is integral over its centre, and is Auslander-Gorenstein. If
R is a prime ring, then the centre
Z(
R) of
R is a Krull domain and
R equals its trace ring TR. Moreover, hd(
R/
M) = height(
M), for every maximal ideal
M of
R.</description><issn>0021-8693</issn><issn>1090-266X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1994</creationdate><recordtype>article</recordtype><recordid>eNp1jzFLAzEYhoMoWKur84063Pkll6TJKEXbQtEiCm4hl3ypKW1TkkPw39ujrk7v9Lw8DyG3FBoKIB82tssN1Zo3lMnJGRlR0FAzKT_PyQiA0VpJ3V6Sq1I2AJQKrkZEztMubdM6OrutVjkdMPcRS5VCdTfL1qO_r14S9l-Yo91Xq0X1Fvfrck0ugt0WvPnbMfl4fnqfzuvl62wxfVzWrmWyr2UISigPbUcFcCl4cIELjrxrxdGZOumo9FpPhAsauRWSgfeoFKNMQKfaMWlOvy6nUjIGc8hxZ_OPoWCGajNUm6HaDNVHQJ0APFp9R8ymuIh7hz5mdL3xKf6H_gJ521yP</recordid><startdate>19940915</startdate><enddate>19940915</enddate><creator>Stafford, J.T.</creator><creator>Zhang, J.J.</creator><general>Elsevier Inc</general><scope>6I.</scope><scope>AAFTH</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>19940915</creationdate><title>Homological Properties of (Graded) Noetherian PI Rings</title><author>Stafford, J.T. ; Zhang, J.J.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c326t-6ff858d03b1504654fcf454e4b351001c6c16d9975cf9e4a5620dde8821250b83</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1994</creationdate><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Stafford, J.T.</creatorcontrib><creatorcontrib>Zhang, J.J.</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>CrossRef</collection><jtitle>Journal of algebra</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Stafford, J.T.</au><au>Zhang, J.J.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Homological Properties of (Graded) Noetherian PI Rings</atitle><jtitle>Journal of algebra</jtitle><date>1994-09-15</date><risdate>1994</risdate><volume>168</volume><issue>3</issue><spage>988</spage><epage>1026</epage><pages>988-1026</pages><issn>0021-8693</issn><eissn>1090-266X</eissn><abstract>Let
R be a connected, graded, Noetherian PI ring. If injdim(
R) =
n < ∞, then we prove that
R is Auslander-Gorenstein and Cohen-Macaulay, with Gelfand-Kirillov dimension equal to
n. If gldim(
R) =
n < ∞, then
R is a domain, finitely generated as a module over its centre and a maximal order in its quotient division ring. Similar results hold if
R is assumed to be local rather than connected graded. Alternatively, suppose that
R is a Noetherian PI ring with gldim(
R) < ∞ such that hd(
R/
M
1) = hd(
R/
M
2) for any two maximal ideals
M
i
in the same clique. Then,
R is a direct sum of prime rings, is integral over its centre, and is Auslander-Gorenstein. If
R is a prime ring, then the centre
Z(
R) of
R is a Krull domain and
R equals its trace ring TR. Moreover, hd(
R/
M) = height(
M), for every maximal ideal
M of
R.</abstract><pub>Elsevier Inc</pub><doi>10.1006/jabr.1994.1267</doi><tpages>39</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | Journal of algebra, 1994-09, Vol.168 (3), p.988-1026 |
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| language | eng |
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| source | Access via ScienceDirect (Elsevier); EZB-FREE-00999 freely available EZB journals |
| title | Homological Properties of (Graded) Noetherian PI Rings |
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