Prime ideals in a large class of nonassociative rings

Author:
Paul J. Zwier

Journal:
Trans. Amer. Math. Soc. **158** (1971), 257-271

MSC:
Primary 17.10

DOI:
https://doi.org/10.1090/S0002-9947-1971-0281763-1

MathSciNet review:
0281763

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper a definition is given for a prime ideal in an arbitrary nonassociative ring under the single restriction that for a given positive integer , if is an ideal in , then is also an ideal. ( is called an -naring.) This definition is used in two ways. First it is used to define the prime radical of and the usual theorems ensue. Second, under the assumption that the -naring has a certain property , the Levitzki radical of is defined and it is proved that is the intersection of those prime ideals in whose quotient rings are Levitzki semisimple. has property if and only if for each finitely generated subring and each positive integer , there is an integer such that . (Here and .)

Furthermore, conditions are given on the identities an -naring satisfies which will insure that satisfies . It is then shown that alternative rings, Jordan rings, and standard rings satisfy these conditions.

**[1]**A. A. Albert,*Power-associative rings*, Trans. Amer. Math. Soc.**64**(1948), 552-593. MR**10**, 349. MR**0027750 (10:349g)****[2]**S. A. Amitsur,*A general theory of radicals*. II, III, Amer. J. Math.**76**(1954), 100-136. MR**15**, 499. MR**0059256 (15:499b)****[3]**B. Brown and N. H. McCoy,*Prime ideals in nonassociative rings*, Trans. Amer. Math. Soc.**89**(1958), 245-255. MR**20**#3196. MR**0096713 (20:3196)****[4]**N. J. Divinsky,*Rings and radicals*, Univ. of Toronto Press, Toronto, 1965. MR**0197489 (33:5654)****[5]**J. Levitzki,*Prime ideals and their lower radical*, Amer. J. Math.**73**(1951), 25-29. MR**12**, 474. MR**0038953 (12:474c)****[6]**R. D. Schafer,*Generalized standard algebras*, J. Algebra**12**(1969), 386-417. MR**0283035 (44:268)****[7]**M. Slater,*Alternative rings with d.c.c.*I, J. Algebra**11**(1969), 102-110. MR**38**#2184. MR**0233863 (38:2184)****[8]**C. Tsai,*The prime radical in Jordan rings*, Proc. Amer. Math. Soc.**19**(1968), 1171-1175. MR**37**#6336. MR**0230776 (37:6336)****[9]**-,*The Levitzki radical in Jordan rings*, Proc. Amer. Math. Soc.**24**(1970), 119-123. MR**0252465 (40:5685)****[10]**A. Thedy,*Zum Wedderburnschen Zerlegungssatz*, Math. Z.**113**(1970), 175-195. MR**0263887 (41:8486)****[11]**K. A. Ževlakov,*Solvability and nilpotence of Jordan rings*, Algebra i Logika Sem.**5**(1966), no. 3, 37-58 (Russian). MR**34**#7601. MR**0207786 (34:7601)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
17.10

Retrieve articles in all journals with MSC: 17.10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0281763-1

Keywords:
-naring,
prime ideal,
semiprime ideal,
prime radical,
Levitzki radical,
-system,
-system,
-radical,
-radical,
alternative ring,
standard ring,
generalized standard ring

Article copyright:
© Copyright 1971
American Mathematical Society