Skew polynomial rings over σ-skew Armendariz rings

Abstract: This article concerns skew polynomial rings over Armendariz rings and skew Armendariz ring. Let R be a Noetherian, Armendariz, prime ring. In this paper we prove that R and the polynomial ring R[x] are 2-primal. Further we prove that if is an endomorphism of a ring R, then (1) R is a -skew Armendariz ring implies that R[x; ] is a -skew Armendariz ring, where is an extension of to R[x; ]. (2) R is a -rigid implies that R[x; ] is a 2-primal.


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One of the earliest examples in non-commutative algebra was skew polynomial rings also known as Ore extensions. Skew polynomial rings have invited attention from Mathematicians and in this area considerable work has been done and investigations are on.
The characterization of ideals and prime ideals (in particular associated prime ideals, completely prime ideals and minimal prime ideals), and 2-primal property of Ore extensions has lead to the extension of certain notions from commutative setup to non-commutative setup. Ore extensions constitute an important class of rings, appearing in extensions of differential calculus, in noncommutative geometry, in quantum groups and algebras and as a uniting framework for many algebras appearing in physics and engineering models.
1993 for more details). An ideal I of a ring R is called completely semi-prime if a 2 ∈ I implies a ∈ I for a ∈ R. We also note that any reduced ring is 2-primal and a commutative ring is also 2-primal.
is not a reduced ring as 2 1 0 2 is a non-zero nilpotent element and hence R is not 2-primal. Krempa (1996) has investigated the relation between minimal prime ideals and completely prime ideals of a ring R. With this he proved the following: Theorem 1.3 For a ring R the following conditions are equivalent: (1) R is reduced.
(2) R is semiprime and all minimal prime ideals of R are completely prime.
(3) R is a subdirect product of domains.
According to Krempa (1996) an endomorphism of a ring R is called rigid if a (a) = 0 implies that a = 0 for a ∈ R. We call a ring -rigid if there exists a rigid endomorphism of R.
Thus R is a -rigid ring.
Example 1.5 (Bhat, 2011, Example 1 Hence R is not a -rigid ring. Also R is said to be -compatible if for each a, b ∈ R, ab = 0 implies and is implied by a (b) = 0.
where u is a fixed element of D. Then R is -compatible.
Example 1.7 Let R = ℤ 2 ⊕ ℤ 2 be a commutative ring, where ℤ 2 is the ring of integers modulo 2. Let :R → R be defined by Then is an automorphism of R.

Armendariz rings
The notion of Armendariz rings was introduced by Rege and Chhawchharia (1997). They defined a ring R to be an Armendariz ring if whenever polynomial (The converse is always true.) This ring was named so because as Armendariz (1974, Lemma 2) had noted that a reduced ring satisfies this condition. In addition to reduced rings, quotient rings over a commutative P.I.D. are Armendariz (Rege & Chhawchharia, 1997, Theorem 2.2). But every n × n full matrix ring over any ring is not Armendariz, where n ≥ 2 (Rege & Chhawchharia, 1997). Note that Armendariz rings are defined through polynomial rings over them. Also subrings of Armendariz rings are Armendariz. Anderson and Camillo (1998) has found a relation between an Armendariz ring and reduced ring as: Proof By Theorem (2.1), R is a reduced ring. We know that a reduced ring is 2-primal. Hence R is 2-primal. ✷ The converse is not true.
Theorem 2.4 Let R be a Noetherian prime ring. If R is an Armendariz ring, then P(R) is completely semi-prime.
Concerning polynomial rings over some kinds of rings, we have the following results: (

-skew Armendariz rings
Recall that R[x; ] is the usual polynomial ring with coefficients in R, in which multiplication is subject to the relation xa = (a)x for all a ∈ R. We take any f (x) ∈ R[x; ] to be of the form f (x) = ∑ n i=0 a i x i . By Anderson and Camillo (1998, Theorem 2), polynomial rings over Armendariz rings are also Armendariz. There is a natural motivation to investigate the nature of skew polynomial ring over a Armendariz ring, but the fact is that a skew polynomial ring over an Armendariz ring need not be Armendariz as follows: Example 3.1 (Kim & Lee, 2000, Example 6) Let ℤ 2 be the ring of integers modulo 2 and consider the ring R = ℤ 2 ⊕ ℤ 2 with the usual addition and multiplication. Then R is a commutative reduced ring; hence R is Armendariz by (1974, Lemma 1). Now let :R → R be defined by Then is an automorphism of R. We claim that R[x; ] is not Armendariz. Let ] is not an Armendariz ring.
We now discuss -skew Armendariz rings ( an endomorphism of a ring R) and their extensions. Recall (Hong et al, 2003) that a ring R with an endomorphism is called a -skew Armendariz ring if for pq = 0 implies that a i i (b j ) = 0, for all 0 ≤ i ≤ m and 0 ≤ j ≤ n. It is also known as skew Armendariz ring with endomorphism . Every subring of a -skew Armendariz ring is -skew Armendariz.
Example 3.2 (Hong et al., 2003, Example 1) Consider the commutative ring Let be an automorphism of R defined by Then R is a -skew Armendraiz ring. Let be an endomorphism of ring R. Then can be extended to an endomorphism (say ) of R[x; ] by We now prove the following Theorem:  With this we prove the following: Theorem 3.6 Let be an endomorphism of ring R such that R is a -rigid. Then R[x; ] is a 2-primal.
Proof By Theorem (3.5), R[x; ] is reduced and hence it is 2-primal. ✷ The converse is not true.