Received 5 May 2016; accepted 17 June 2016; published 20 June 2016

1. Introduction
Throughout this paper every ring is an associative with identity unless otherwise stated. Given a ring R,
,
,
,
,
and
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
matrix ring over R, the ring of integers modulo n, respectively. A ring
is called Skew polynomial ring if
is an endomorphism over R; operations are usual addition and multiplication defined by
. In [1] , that skew polynomial ring has no Morita invariance. A ring R is called Armendariz ring if
implies
, where
,
for any
,
in [2] . If R is a semiprime ring, then skew polynomial ring
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.
2. Preliminary
Definition 2.1. [6] Let R and S be rings, then R and S are Morita equivalent if there exists projective module
, such that
. 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
and
.
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
is a monomorphism over R, then skew polynomial ring
has no zero divisor.
Proof. For any
,
, if
![]()
then all coefficients of the skew polynomial are zero. Since
is a monomorphism and R has no zero divisor, so
implies
,
implies
or
. Case 1. If
,
, then
. Since
, so
. Simlarly, we show
, thus
. Case 2. If
,
, then
. Because
, so
,
. Similarly, ![]()
, thus
. Case 3. If
,
, so
,
. Similarly,
or
for
,
, then
or
.Therefore
is a ring has no divisor of zero.
Definition 2.5. [8] A ring R is called a reversible, if
implies
for any
.
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
are the right annihilator of all coefficients of the polynomial.
Proof. For any
,
, let
, then
, so
,
,
,
,
and
. Because R be a reduced ring, hence R be a reversible ring, so
. Because
, then
, so
, so
, thus
and
, so
, so
and
. Since
, we have
, then
, so
, so
and
, so
, so
, hence
, so
,
, so
, thus
, Similarly, we have
, which
,
. Therefore the coefficients of right annihilator of any polynomial over
are the right annihilator of all coefficients of the polynomial.
Proposition 2.8. Let R be a reduced ring, then the idempotent of ring R is the idempotent of
.
Proof. For any
, if
, we have
![]()
then
. Since R is a reduced ring and
,
, so
,
. Similarly, we have
, and because
, so
. Thus
, which
is the idempotent of
.
3. Main Results
The property of skew polynomial ring relation to Morita invariance , we have the following counterexample.
Example 3.1. Suppose that a ring
and
is an endomorphism over
, define the usual addition and multiplication by
for any
, then
is a skew polynomial ring, but has no Morita invariance.
In fact, clearly,
is a skew polynomial ring. Consider a ring
, which
,
,
,
, we have all idempotents of
are
,
,
,
,
,
. Suppose that
.
Case 1. If
, then
![]()
Case 2. If
, then
. Case 3. If
, then
. Similarly, we show that
for all idempotente. Clearly, the condition of
is not true for any integer n, so
and any ring S are not Morita equivalent by lemma 2.2, therefore
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
be a monomorphism over R, then skew polynomial ring
is an Armendariz ring.
Proof. Since R has no zero divisor, so
has no zero divisor by lemma2.4, then
is a reversible ring. For any
,
, and
,
, if
![]()
then the all coefficients of
are zero. Since
,
, so
, and hence
,
. Because
,
, so
,
, and have
,
. Similarly,
. Thus the skew polynomial ring
of no zero divisor is an Armendariz ring.
Next research the necessary and sufficient of this condition by the following example.
Example 3.3. Let
be a ring with a monomorphism
defined by
. For any
,
, define the usual addition and multiplication by
, then
is a skew polynomial ring, but
is not an Armendariz ring.
In fact, clearly,
is a skew polynomial ring. Let
,
, if
, then all coefficients of the skew polynomial
are zero. But
,
, thus
is not an Armendariz ring.
It derives from the above example 3.3 that we further verify the condition is necessary. Next we study that skew polynomial ring
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
is an endomorphism over R, then skew polynomial ring
is a (quasi)-Baer ring.
Proof. For any
,
, let
. If
, then
is any polynomial ring, and has
. If
, there exists
, such that
, then
. And because R is a ring has no ze-
ro divisor, so
or
. Since arbitrary of
and i, we have
implies
, so
. Thus the right annihilators set of any nonempty subset X is
. So
is a Baer ring, and
is a quasi-Baer ring by [5] .
The following example shows that skew polynomial ring is (quasi)-Baer ring.
Example 3.5. Let
be a ring with an endomorphism
defined by
, then
is a skew polynomial ring, and
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
is
, then the right ideal generated by the idempotent 0. Thus
is a (quasi)-Baer ring clearly by theorem 3.4.
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.
Example 3.6. Suppose that
be a ring with an endomorphism
defined by
over R, for any
,
, define the usual addition and multiplication is defined by
, then
is a skew polynomial ring, but is not a (quasi)-Baer ring.
In fact, clearly,
is a skew polynomial ring. For any
, if
, then
,
,
,
.We have all idempotents of
are
,
,
,
. Let
, since
![]()
which
. Thus
is not a (quasi)-Baer ring.
4. Conclusion
Inthis paper, we show that skew polynomial ring has no Morita invariance by the counterexample, and give the condition that skew polynomial ring constitutes Armendariz and (quasi)-Baer ring, and verify that the condition is necessary, but not sufficient.
Acknowledgements
The authors thank the referee for very careful reading the manuscript and many valuable suggestions that improved the paper by much. This work was supported by the National Natural Science Foundation of China (11361063).
NOTES
![]()
*Corresponding author.