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Zero divisors and finite near-rings

Published online by Cambridge University Press:  09 April 2009

S. Ligh  and
J. J. Malone Jr
S. Ligh
Affiliation:
Department of Mathematics, Texas A & M University College Station, Texas 77843, United States of America
J. J. Malone Jr
Affiliation:
Department of Mathematics, Texas A & M University College Station, Texas 77843, United States of America
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A near-ring is a triple (R, +, · such that (R, +) is a group, (R, ·) is a semigroup, and is left distributive work on near-rings is [1]. A near-ring R is distributively generated if there exists S ⊂ R such that (S, ·) is sub-semigroup of (R, ·), each element of S is right distributive, and S is an additive generating set for (R, +). Distributively generated near-rings, first treated in [3], arise out of consideration of the system generated by the endomorphisms of an (not necessarily commutative) additive group. A near-field is a near-ring such that the nozero elements form a group under multiplication. Near fields are discussed in [9]. An element x ≠ 0 in R is a left (right) zero divisor if there is a ≠ 0 in R such that xa = 0 (ax = 0). A zero divisor is an element that is either a left or a right zero divisor. In a near-ring R it will be assumed that Ox = 0 for each x ∈ R.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 11 , Issue 3 , August 1970 , pp. 374 - 378
Copyright
Copyright © Australian Mathematical Society 1970

References

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