Polynomial rings over Goldie-Kerr commutative rings

Author:
Carl Faith

Journal:
Proc. Amer. Math. Soc. **120** (1994), 989-993

MSC:
Primary 13E10; Secondary 13B25, 16P60

DOI:
https://doi.org/10.1090/S0002-9939-1994-1221723-8

MathSciNet review:
1221723

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Abstract: All rings in this paper are commutative, and (resp., ) denotes the acc on annihilators (resp., on direct sums of ideals). Any subring of an ring, e.g., of a Noetherian ring, is an ring. Together, and constitute the requirement for a ring to be a Goldie ring. Moreover, a ring is Goldie iff its classical quotient ring is Goldie.

A ring is a Kerr ring (the appellation is for J. Kerr, who in 1990 constructed the first Goldie rings not Kerr) iff the polynomial ring has (in which case must have ). By the Hilbert Basis theorem, if is a Noetherian ring, then so is ; hence, any subring of a Noetherian ring is Kerr.

In this note, using results of Levitzki, Herstein, Small, and the author, we show that any Goldie ring such that has nil Jacobson radical (equivalently, the nil radical of is an intersection of associated prime ideals) is Kerr in a very strong sense: is Artinian and, hence, Noetherian (Theorems 1.1 and 2.2). As a corollary we prove that any Goldie ring that is algebraic over a field is Artinian, and, hence, any order in is a Kerr ring (Theorem 2.5 and Corollary 2.6). The same is true of any algebra over a field of cardinality exceeding the dimension of (Corollary 2.7).

Other Kerr rings are: reduced rings and valuation rings with (see 3.3 and 3.4).

**[C]**Victor P. Camillo,*Commutative rings whose quotients are Goldie*, Glasgow Math. J.**16**(1975), no. 1, 32–33. MR**0379476**, https://doi.org/10.1017/S0017089500002470**[CG]**Victor Camillo and Robert Guralnick,*Polynomial rings over Goldie rings are often Goldie*, Proc. Amer. Math. Soc.**98**(1986), no. 4, 567–568. MR**861751**, https://doi.org/10.1090/S0002-9939-1986-0861751-0**[F1]**Carl Faith,*Finitely embedded commutative rings*, Proc. Amer. Math. Soc.**112**(1991), no. 3, 657–659. MR**1057942**, https://doi.org/10.1090/S0002-9939-1991-1057942-0**[F2]**-,*Annihilators, associated prime ideals, and Kasch-McCoy quotient rings of commutative rings*, Comm. Algebra**119**(1991), 1867-1892.**[F-P]**Carl Faith and Poobhalan Pillay,*Classification of commutative FPF rings*, Notas de Matemática [Mathematical Notes], vol. 4, Universidad de Murcia, Secretariado de Publicaciones e Intercambio Científico, Murcia, 1990. MR**1091714****[GM]**S. M. Ginn and P. B. Moss,*Finitely embedded modules over Noetherian rings*, Bull. Amer. Math. Soc.**81**(1975), 709–710. MR**0369424**, https://doi.org/10.1090/S0002-9904-1975-13831-6**[HP]**Dolors Herbera and Poobhalan Pillay,*Injective classical quotient rings of polynomial rings are quasi-Frobenius*, J. Pure Appl. Algebra**86**(1993), no. 1, 51–63. MR**1213153**, https://doi.org/10.1016/0022-4049(93)90152-J**[HS]**I. N. Herstein and L. Small,*Nil rings satisfying certain chain conditions*, Canad. J. Math.**16**(1964), 771–776. MR**0166220**, https://doi.org/10.4153/CJM-1964-074-0**[J]**N. Jacobson,*The structure of rings*, Amer. Math. Soc., Providence, RI, 1956; revised ed. 1964.**[K1]**Jeanne Wald Kerr,*The polynomial ring over a Goldie ring need not be a Goldie ring*, J. Algebra**134**(1990), no. 2, 344–352. MR**1074333**, https://doi.org/10.1016/0021-8693(90)90057-U**[K2]**Jeanne Wald Kerr,*An example of a Goldie ring whose matrix ring is not Goldie*, J. Algebra**61**(1979), no. 2, 590–592. MR**559857**, https://doi.org/10.1016/0021-8693(79)90297-7**[La]**Joachim Lambek,*Lectures on rings and modules*, 2nd ed., Chelsea Publishing Co., New York, 1976. MR**0419493****[Le]**J. Levitzki,*On nil subrings*, Israel J. Math.**1**(1963), 215–216. MR**0163931**, https://doi.org/10.1007/BF02759721**[P]**C. Perelló (ed.),*Pere Menal Memorial volumes*, Publ. Mat.**36**(2A & 2B) (1992).**[R]**Moshe Roitman,*On polynomial extensions of Mori domains over countable fields*, J. Pure Appl. Algebra**64**(1990), no. 3, 315–328. MR**1061306**, https://doi.org/10.1016/0022-4049(90)90065-P**[S1]**Lance W. Small,*Orders in Artinian rings*, J. Algebra**4**(1966), 13–41. MR**0200300**, https://doi.org/10.1016/0021-8693(66)90047-0**[S2]**Lance W. Small,*The embedding problem for Noetherian rings*, Bull. Amer. Math. Soc.**75**(1969), 147–148. MR**0232797**, https://doi.org/10.1090/S0002-9904-1969-12183-X

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DOI:
https://doi.org/10.1090/S0002-9939-1994-1221723-8

Article copyright:
© Copyright 1994
American Mathematical Society