Kurosh‐Amitsur Right Jacobson Radical of Type 0 for Right Near‐Rings

Abstract

By a near-ring we mean a right near-ring.J0r, the right Jacobson radical of type 0, was introduced for near-rings by the first and second authors. In this paper properties of the radicalJ0rare studied. It is shown thatJ0ris a Kurosh-Amitsur radical (KA-radical) in the variety of all near-ringsR, in which the constant partRcofRis an ideal ofR. So unlike the left Jacobson radicals of types 0 and 1 of near-rings,J0ris a KA-radical in the class of all zero-symmetric near-rings.J0ris nots-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings.

Special Right Jacobson Radicals for Right Near-rings

“…Therefore, P is a right g ν -primitive ideal of R. Proof. Let h 0 be a generator of the right S-group G. From the proof of Theorem 3.10 of [9], for h ∈ H, r ∈ R the operation defined by hr := h 0 (sr) if h = h 0 s, s ∈ S, makes G a right R-group and is an extension the action of G on S to R. Moreover, Theorem 3.10 of [9] and Theorems 3.9 and 3.10 of [10], G is a right R-group of type-ν, for ν ∈ {1, 2}. Since G is a right R-group of type-ν(e), by Theorem 3.…”

“…Since G is a faithful right R-group, (0 : G) R := {r ∈ R | Gr = {0}} = {0}. From the proof of Theorem 3.10 of [9], it can be easily seen that a generator of the right S-group G is also a generator of the right R-group G. So A is the set of generators of the right R-group G. Suppose that r ∈ (0 : A). Now Ar = {0}.…”

“…Srinivasa Rao and Siva Prasad [6,7] introduced and studied J r ν , the right Jacobson radical type-ν, ν ∈ {0, 1, 2}. In [9,10] Srinivasa Rao and Siva Prasad along with T. Srinivas showed that J r ν is a Kurosh-Amitsur radical in the Fuchs variety F of all near-rings R in which the constant part R c of R is an ideal of R, ν ∈ {0, 1, 2}. But J r ν is not s-hereditary in the class of all zero-symmetric near-rings and hence it is not an ideal-hereditary radical in that class, ν ∈ {0, 1, 2}.…”

“…Then G is a right S-group of type-g ν , ν ∈ {0, 1, 2}.Proof. If G is a right R-group of type-0 and S is an invariant subnear-ring and right ideal of R with GS ̸ = {0}, then under the restriction of G to S, by Theorem 3.2 of[9], G is a right S-group type-0. Also if G be a right R-group of type-ν and S is an invariant subnear-ring of R with GS ̸ = {0}, then under the restriction of G to S, by Theorems 3.1 and 3.2 of[10], G is a right S-group type-ν, where ν ∈ {1, 2}.…”

“…Let A be the set of generators of the right R-group G and P be the largest ideal of R contained in (0 : G) R := {r ∈ R | Gr = {0}}. A generator of the right R-group G is also a generator of the right S-group G. From the proof of Theorem 3.10 of[9] (and Theorems 3.9 and 3.10 of[10] for ν ∈ {1, 2}) as the extension of G from S to R coincides with the action of G on R, it follows that a generator of the right S-group G is also a generator of the right R-group G. So A is the set of generators of the right S-group G. We have P = (0 : A) = {r ∈ R | ar = 0 for all a ∈ A}. NowP ∩ S = (0 : A) ∩ S = {s ∈ S | As = {0}}.…”