Abstract
It is well-known that every semisimple class in the class of all associative rings or alternative rings is hereditary, this being an easy consequence of the Anderson-Divinsky-Suliński property (cf. [1]). Dropping the associativity, one get degenerate radicals in the sense that a radical class has a hereditary semisimple class if and only if it is an A-radical, i.e. the radical only depends on the structure of the underlying abelian groups (cf. [4]a and [5]). In the class of all near-rings (or abelian near-rings) the situation is not as bad, here we have both hereditary and non-hereditary semisimple classes, see, for example [3]. Dropping the associativity in this case, the result is even worse than in that of non-associative rings. We show that in the class of all abelian, not-necessatily associative zero-symmetric near-rings the only radicals with hereditary semisimple classes are the two trivial radical classes.
| Original language | English |
|---|---|
| Pages (from-to) | 273-275 |
| Number of pages | 3 |
| Journal | North-Holland Mathematics Studies |
| Volume | 137 |
| Issue number | C |
| DOIs | |
| Publication status | Published - 1987 |
Fingerprint
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Some Pathology for Radicals in Non-Associative Near-Rings. / Veldsman, Stefan.
In: North-Holland Mathematics Studies, Vol. 137, No. C, 1987, p. 273-275.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Some Pathology for Radicals in Non-Associative Near-Rings
AU - Veldsman, Stefan
PY - 1987
Y1 - 1987
N2 - It is well-known that every semisimple class in the class of all associative rings or alternative rings is hereditary, this being an easy consequence of the Anderson-Divinsky-Suliński property (cf. [1]). Dropping the associativity, one get degenerate radicals in the sense that a radical class has a hereditary semisimple class if and only if it is an A-radical, i.e. the radical only depends on the structure of the underlying abelian groups (cf. [4]a and [5]). In the class of all near-rings (or abelian near-rings) the situation is not as bad, here we have both hereditary and non-hereditary semisimple classes, see, for example [3]. Dropping the associativity in this case, the result is even worse than in that of non-associative rings. We show that in the class of all abelian, not-necessatily associative zero-symmetric near-rings the only radicals with hereditary semisimple classes are the two trivial radical classes.
AB - It is well-known that every semisimple class in the class of all associative rings or alternative rings is hereditary, this being an easy consequence of the Anderson-Divinsky-Suliński property (cf. [1]). Dropping the associativity, one get degenerate radicals in the sense that a radical class has a hereditary semisimple class if and only if it is an A-radical, i.e. the radical only depends on the structure of the underlying abelian groups (cf. [4]a and [5]). In the class of all near-rings (or abelian near-rings) the situation is not as bad, here we have both hereditary and non-hereditary semisimple classes, see, for example [3]. Dropping the associativity in this case, the result is even worse than in that of non-associative rings. We show that in the class of all abelian, not-necessatily associative zero-symmetric near-rings the only radicals with hereditary semisimple classes are the two trivial radical classes.
UR - http://www.scopus.com/inward/record.url?scp=77956896880&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=77956896880&partnerID=8YFLogxK
U2 - 10.1016/S0304-0208(08)72310-7
DO - 10.1016/S0304-0208(08)72310-7
M3 - Article
VL - 137
SP - 273
EP - 275
JO - North-Holland Mathematics Studies
JF - North-Holland Mathematics Studies
SN - 0304-0208
IS - C
ER -