The Some Properties of Skew Polynomial Rings

This paper mainly studies some properties of skew polynomial ring related to Morita invariance, Armendariz and (quasi)-Baer. First, we show that skew polynomial ring has no Morita invariance by the counterexample. Then we prove a necessary condition that skew polynomial ring constitutes Armendariz ring. We lastly investigate that condition of skew polynomial ring is a (quasi)-Baer ring, and verify that the conditions is necessary, but not sufficient by example and counterexample.


Introduction
Throughout this paper every ring is an associative with identity unless otherwise stated.Given a ring R, [ ]

( )
Mat n R and n  denote the polynomial ring with an indeterminate x over R, the skew polynomial ring over R, the right annihilator of nonempty subset X of ring R, the left annihilator of nonempty subset X of ring R, and the n n × matrix ring over R, the ring of integers modulo n, respectively.A ring .In [1], that skew polynomial ring has no Morita invariance.A ring R is called Armendariz ring if ( ) ( ) 0 f x g x = implies 0 i j a b = , where 0 i m ≤ ≤ , 0 j n ≤ ≤ for any ( ) ∑ in [2].If R is a semiprime ring, then skew polynomial ring

[ ]
; R x σ is a quasi-Armendariz ring by [3].G. F. Birkenmeier first introduced the concept of Baer ring, and proved that Baer ring is quasi-Baer ring, but converse is not hold, and right principally quasi-Baer ring has Morita invariance by [4].Q.J. Song gave the condition that iterated skew polynomial ring constitutes (quasi)-Baer ring by [5].We will show that skew polynomial ring has no Morita invariance by the counterexample, and the condition that skew polynomial ring has properties of Armendarizand (quasi)-Baer, and verify that the condition is necessary, but not sufficient by example and counterexample.

Preliminary
Definition 2.1.[6] Let R and S be rings, then R and S are Morita equivalent if there exists projective module . Morita invariance is the invariant property under Morita equivalent rings.Lemma 2.2.[6] The ring R and S are Morita equivalent, if and only if there exists an integer n and idempotent ( ) , such that ( ) . Definition 2.3.[7] A ring R is called (quasi)-Baer ring if the right annihilator of (resp.right ideal) nonempty subset of R is generated by an idempotent as a right ideal.
Lemma 2.4.Suppose that R is a ring has no zero divisor and σ is a monomorphism over R, then skew po- Proof.For any ( )  then all coefficients of the skew polynomial are zero.Since σ is a monomorphism and R has no zero divisor, so Proof.For any ( ) Proposition 2.8.Let R be a reduced ring, then the idempotent of ring R is the idempotent of [ ] R x .

Main Results
The property of skew polynomial ring relation to Morita invariance , we have the following counterexample.) Mat Z ; , which ( ) Mat Z ; x σ Mat Z ; Mat Z ; Mat Z ; x σ has no Morita invariance.
So the skew polynomial ring has no Morita invariance by the counterexample.The following theorem shows that the condition of skew polynomial ring constitutes Armendariz ring.
Theorem 3.2.Let R be a ring that has no zero divisor and σ be a monomorphism over R, then skew poly- . Thus the skew polynomial ring [ ] ; R x σ of no zero divisor is an Armendariz ring.Next research the necessary and sufficient of this condition by the following example.
be a ring with a monomorphism σ defined by 0 0 For any ( ) , define the usual addition and multiplication by ( ) In fact, clearly, [ ] ; R x σ is a skew polynomial ring.Let ( ) , then all coefficients of the skew polynomial [ ] It derives from the above example 3.3 that we further verify the condition is necessary.Next we study that skew polynomial ring [ ] ; R x σ is a (quasi)-Baer ring under the condition of no zero divisor .Theorem 3.4.Let R be a ring that has no zero divisor and σ is an endomorphism over R, skew poly- nomial ring [ ] ; R x σ is a (quasi)-Baer ring.
Proof.For any ( ) ( ) g x is any polynomial ring, and has [ ] ( ) So we proof the condition of no zero divisor is necessary.The following counterexample shows that the condition is not sufficient condition that skew polynomial ring is a (quasi)-Baer ring.
is called Skew polynomial ring if σ is an endomorphism over R; operations are usual addition and multiplication defined by

Definition 2 . 5 .
is a ring has no divisor of zero.[8]A ring R is called a reversible, if 0 ab = implies 0 ba = for any , a b R ∈ .Proposition 2.6.[9]Every reduced ring is a reversible ring, but the converse does not hold.Proposition 2.7.Let R be a reduced ring, then the coefficients of right annihilator of any polynomial over [ ] R x are the right annihilator of all coefficients of the polynomial.
Therefore the coefficients of right annihilator of any polynomial over [ ] R x are the right annihilator of all coefficients of the polynomial.

Example 3 . 1 .
Suppose that a ring 2 Z and σ is an endomorphism over 2 Z , define the usual addition and multiplication by nomial ring [ ] ; R x σ is an Armendariz ring.Proof.Since R has no zero divisor, so [ ] ; R x σ has no zero divisor by lemma2.4,then [ ] . Thus the right annihilators set of any nonempty subset X is is a Baer ring, and [ ] ; R x σ is a quasi-Baer ring by[5].The following example shows that skew polynomial ring is (quasi)-Baer ring. is a skew polynomial ring, and [ ] ; R x σ is a (quasi)-Baer ring.In fact, clearly, R is a field, so R is a no zero divisor ring.Therefore the right annihilator of every nonempty subset , then the right ideal generated by the idempotent 0. Thus [ ] ; R x σ is a (quasi)-Baer ring clearly by theorem 3.4.
is a skew polynomial ring, but is not a (qua- si)-Baer ring.In fact, clearly, [ ]; R x σ is a skew polynomial ring.For any (