TY - JOUR

T1 - Nearly Finite-Dimensional Jordan Algebras

AU - Zhelyabin, V. N.

AU - Panasenko, A. S.

PY - 2018/11/1

Y1 - 2018/11/1

N2 - Nearly finite-dimensional Jordan algebras are examined. Analogs of known results are considered. Namely, it is proved that such algebras are prime and nondegenerate. It is shown that the property of being nearly finite-dimensional is preserved in passing from an alternative algebra to an adjoint Jordan algebra. A similar result is established for associative nearly finite-dimensional algebras with involution. It is stated that a nearly finite-dimensional Jordan PI-algebra with unity either is a finite module over a nearly finite-dimensional center or is a central order in an algebra of a nondegenerate symmetric bilinear form. Also the following result holds: if a locally nilpotent ideal has finite codimension in a Jordan algebra with the ascending chain condition on ideals, then that algebra is finite-dimensional. In addition, E. Formanek’s result in [Comm. Alg., 1, No. 1, 79-86 (1974)], which says that associative prime PI-rings with unity are embedded in a free module of finite rank over its center, is generalized to Albert rings.

AB - Nearly finite-dimensional Jordan algebras are examined. Analogs of known results are considered. Namely, it is proved that such algebras are prime and nondegenerate. It is shown that the property of being nearly finite-dimensional is preserved in passing from an alternative algebra to an adjoint Jordan algebra. A similar result is established for associative nearly finite-dimensional algebras with involution. It is stated that a nearly finite-dimensional Jordan PI-algebra with unity either is a finite module over a nearly finite-dimensional center or is a central order in an algebra of a nondegenerate symmetric bilinear form. Also the following result holds: if a locally nilpotent ideal has finite codimension in a Jordan algebra with the ascending chain condition on ideals, then that algebra is finite-dimensional. In addition, E. Formanek’s result in [Comm. Alg., 1, No. 1, 79-86 (1974)], which says that associative prime PI-rings with unity are embedded in a free module of finite rank over its center, is generalized to Albert rings.

KW - Albert ring

KW - associative nearly finite-dimensional algebra with involution

KW - nearly finite-dimensional Jordan algebra

KW - nearly finite-dimensional Jordan PI-algebra with unity

UR - http://www.scopus.com/inward/record.url?scp=85057748269&partnerID=8YFLogxK

U2 - 10.1007/s10469-018-9506-5

DO - 10.1007/s10469-018-9506-5

M3 - Article

AN - SCOPUS:85057748269

VL - 57

SP - 336

EP - 352

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 5

ER -