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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.

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

Definition 2.1. [

Lemma 2.2. [

Definition 2.3. [

Lemma 2.4. Suppose that R is a ring has no zero divisor and

Proof. For any

then all coefficients of the skew polynomial are zero. Since

Definition 2.5. [

Proposition 2.6. [

Proposition 2.7. Let R be a reduced ring, then the coefficients of right annihilator of any polynomial over

Proof. For any

Proposition 2.8. Let R be a reduced ring, then the idempotent of ring R is the idempotent of

Proof. For any

then

The property of skew polynomial ring relation to Morita invariance , we have the following counterexample.

Example 3.1. Suppose that a ring

In fact, clearly,

Case 1. If

Case 2. If

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

Proof. Since R has no zero divisor, so

then the all coefficients of

Next research the necessary and sufficient of this condition by the following example.

Example 3.3. Let

In fact, clearly,

It derives from the above example 3.3 that we further verify the condition is necessary. Next we study that skew polynomial ring

Theorem 3.4. Let R be a ring that has no zero divisor and

Proof. For any

ro divisor, so

The following example shows that skew polynomial ring is (quasi)-Baer ring.

Example 3.5. Let

In fact, clearly, R is a field, so R is a no zero divisor ring. Therefore the right annihilator of every nonempty subset

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

In fact, clearly,

which

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.

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).

Qianqian Chu,Hailan Jin, (2016) The Some Properties of Skew Polynomial Rings. Advances in Pure Mathematics,06,507-511. doi: 10.4236/apm.2016.67037