### 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 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*North-Holland Mathematics Studies*,

*137*(C), 273-275. https://doi.org/10.1016/S0304-0208(08)72310-7

**Some Pathology for Radicals in Non-Associative Near-Rings.** / Veldsman, Stefan.

Research output: Contribution to journal › Article

*North-Holland Mathematics Studies*, vol. 137, no. C, pp. 273-275. https://doi.org/10.1016/S0304-0208(08)72310-7

}

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 -