7_bhat.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 6, No. 1, 2013, 59-65

ISSN 1307-5543 – www.ejpam.com

Skew Polynomial Rings over Weak σ-rigid Rings and σ(∗)-rings

Neetu Kumari, Smarti Gosani, V. K. Bhat∗

School of Mathematics, SMVD University, P/o SMVD University, Katra, J and K, India- 182320

Abstract. Let R be a ring and σ an endomorphism of R. Recall that R is said to be a σ(∗)-ring if

aσ(a) ∈ P(R) implies a ∈ P(R) for a ∈ R, where P(R) is the prime radical of R. We also recall that R is

said to be a weak σ-rigid ring if aσ(a) ∈ N(R) if and only if a ∈ N(R) for a ∈ R, where N(R) is the set

of nilpotent elements of R.

In this paper we give a relation between a σ(∗)-ring and a weak σ-rigid ring. We also give a necessary

and sufficient condition for a Noetherian ring to be a weak σ-rigid ring. Let σ be an endomorphism of

a ring R. Then σ can be extended to an endomorphism (say σ) of R[x;σ]. With this we show that if R

is a Noetherian ring and σ an automorphism of R, then R is a weak σ-rigid ring if and only if R[x;σ]

is a weak σ-rigid ring.

2010 Mathematics Subject Classifications: 16-XX; 16S36, 16P40, 16P50, 16U20.

Key Words and Phrases: Automorphism, σ(∗)-ring, weak σ-rigid ring, 2-primal ring

1. Introduction

A ring R always means an associative ring with identity 1 6= 0. The ring of integers is

denoted by Z, and the set of positive integers is denoted by N. The set of prime ideals of R is

denoted by Spec(R). The sets of minimal prime ideals of R is denoted by Min.Spec(R). The

prime radical and the nil radical of R are denoted by P(R) and N(R) respectively.

Now let R be a ring and σ an endomorphism of R. Recall that the skew polynomial ring

R[x ;σ] is the set of polynomials

{
n
∑

i=0

x iai, ai ∈ R, n ∈ N}

with usual addition of polynomials and multiplication subject to the relation ax = xσ(a) for

all a ∈ R. We take any f (x) ∈ R[x ;σ] to be of the form f (x) =
∑n

i=0 x iai, n ∈ N as followed

in McConnell and Robson [12]. We denote R[x ;σ] by S(R).

∗Corresponding author.

Email address: vijaykumarbhat2000�yahoo.
om (V. Bhat)

http://www.ejpam.com 59 c© 2013 EJPAM All rights reserved.



N. Kumari, S. Gosani, V. Bhat / Eur. J. Pure Appl. Math, 6 (2013), 59-65 60

Skew-polynomial rings have been of interest to many authors. For example [1, 2, 5, 7, 10,

11, 13].

The classical study of any commutative Noetherian ring is done by studying its primary

decomposition and this forms the fundamental edifice on which any such ring is studied.

Further there are other structural properties of rings, for example the existence of quotient

rings or more particularly the existence of Artinian quotient rings etc. which can be nicely

tied to primary decomposition of a Noetherian ring. The notion of the quotient ring of a ring,

the contractions and extensions of ideals arising thereby appear in Chapter 9 of [7].

The first important result in the theory of non commutative Noetherian rings was proved

in 1958 (Goldie’s Theorem) which gives an analogue of field of fractions for factor rings R/P,

where R is a Noetherian ring and P is a prime ideal of R. In 1959 the one sided version was

proved by Lesieur and Croisot [Theorem 5.12 of 7] and in 1960 Goldie generalized the result

for semiprime rings [Theorem 5.10 of 7].

In [5] it is shown that if R is embeddable in a right Artinian ring and if characteristic of

R is zero, then the differential operator ring R[x ;δ] embeds in a right Artinian ring. It is also

shown in [5] that if R is a commutative Noetherian ring and σ is an automorphism of R, then

the skew-polynomial ring R[x ;σ] embeds in an Artinian ring.

A non commutative analogue of associated prime ideals of a Noetherian ring has also been

discussed. We would like to note that a considerable work has been done in the investiga-

tion of prime ideals (in particular minimal prime ideals and associated prime ideals) of skew

polynomial rings (K. R. Goodearl and E. S. Letzter [8], C. Faith [6], S. Annin [1], Leroy and

Matczuk [11], Nordstrom [13]) and Bhat [2].

Another related area of interest since recent past has been the study of 2-primal rings. This

involves the notions of prime radical and the set of nilpotent elements of a ring. Furthermore

the concept of completely prime ideals and the completely semiprime ideals are also studied

in this area. Krempa in [9] introduced σ-rigid rings; Kwak in [10] introduced σ(∗)-rings and

Ouyang in [14] introduced weak σ-rigid rings, where σ is an endomorphism of ring R. These

rings are related to 2-primal rings. In this paper we study these rings and find a relation

between them. Towards this we prove the following theorem:

Let R be a ring. Let σ be an endomorphism of R such that R is a σ(∗)-ring. Then R is a weak

σ-rigid ring. Conversely a 2-primal weak σ-rigid ring is a σ(∗)-ring. (This is proved in Theorem

2).

We also discuss skew polynomial rings over weak σ-rigid rings.

We note that if σ is an endomorphism of a ring R, then it can be extended to an endomor-

phism σ of S(R) = R[x ;σ] by σ(
∑m

i=0 x iai) =
∑m

i=0 x iσ(ai). With this we prove the following

theorem:

Let R be a Noetherian ring. Let σ be an automorphism of R. Then R is a weak σ-rigid ring if

and only if S(R) = R[x ;σ] is a weak σ-rigid ring. (This is proved in Theorem 3).



N. Kumari, S. Gosani, V. Bhat / Eur. J. Pure Appl. Math, 6 (2013), 59-65 61

2. Preliminaries

We begin with the following definitions:

Definition 1 (Krempa [9]). An endomorphism σ of a ring R is said to be rigid if aσ(a) = 0

implies a = 0 for a ∈ R. A ring R is said to be σ-rigid if there exists a rigid endomorphism σ of

R.

Definition 2 (Kwak [10]). Let R be a ring and σ an endomorphism of R. Then R is said to be a

σ(∗)-ring if aσ(a) ∈ P(R) implies a ∈ P(R) for a ∈ R.

Example 1 (Kwak [10]). Let R =

�

F F

0 F

�

, where F is a field. Then P(R) =

�

0 F

0 0

�

.

Let σ : R → R be defined by σ
�

�

a b

0 c

�

�

=

�

a 0

0 c

�

. Then it can be seen that σ is an

endomorphism of R and R is a σ(∗)-ring.

We note that the above ring is not σ-rigid. Let 0 6= a ∈ F . Then

�

0 a

0 0

�

σ

�

0 a

0 0

�

=

�

0 0

0 0

�

, but

�

0 a

0 0

�

6=

�

0 0

0 0

�

.

In [10], Kwak also establishes a relation between a 2-primal ring and a σ(∗)-ring. Recall

that a ring R is 2-primal if N(R) = P(R). Also an ideal I of a ring R is called completely

semiprime if a2 ∈ I implies a ∈ I for a ∈ R. Clearly R is a I(*)-ring if and only if R is a 2-primal

ring, where I is the identity map on R. The ring in Example 1 is 2-primal.

In [10], the 2-primal property has also been extended to the skew-polynomial ring R[x ;σ].

We now give an example of a ring R, and an endomorphism σ of R such that R is not a

σ(∗)-ring, however R is 2-primal.

Example 2 (Kwak [10]). Let R = F[x] be the polynomial ring over a field F. Then R is 2-primal

with P(R) = 0. Let σ : R→ R be an endomorphism defined by σ( f (x)) = f (0). Then R is not a

σ(∗)-ring. For example consider f (x) = xa, a 6= 0.

Let R be a ring and σ an automorphism of R. We now give a necessary and sufficient

condition for R to be a σ(∗)-ring in the following proposition:

Proposition 1. Let R be a Noetherian ring and σ an automorphism of R. Then R is a σ(∗)-ring

implies that P(R) is completely semiprime.

Proof. Let R be a σ(∗)-ring. We show that P(R) is completely semiprime. Let a ∈ R be such

that a2 ∈ P(R). Then

aσ(a)σ(aσ(a)) = aσ(a)σ(a)σ2(a) ∈ σ(P(R)) = P(R).

Therefore aσ(a) ∈ P(R) and hence a ∈ P(R).

Converse of the above need not be true.



N. Kumari, S. Gosani, V. Bhat / Eur. J. Pure Appl. Math, 6 (2013), 59-65 62

Example 3 (Kwak [10]). Let K be a field, R = K × K and the automorphism σ of R defined by

σ((a, b)) = (b, a), a, b ∈ K. Then R is a reduced ring and so P(R) = 0 is completely semiprime.

But the ring R is not a σ(∗)-ring since (1,0)σ((1,0)) = (0,0) but (1,0) /∈ P(R).

Recall that an ideal P of a ring R is completely prime if ab ∈ P implies a ∈ P or b ∈ P for

a, b ∈ R. In commutative sense completely prime and prime have the same meaning. We also

note that every completely prime ideal of a ring R is a prime ideal, but the converse need not

be true.

The following example shows that a prime ideal need not be a completely prime ideal.

Example 4. Let R=

�

Z Z

Z Z

�

= M2(Z). If p is a prime number, then the ideal P = M2(pZ) is

a prime ideal of R, but is not completely prime, since for a =

�

1 0

0 0

�

and b =

�

0 0

0 1

�

, we

have ab ∈ P, even though a /∈ P and b /∈ P.

There are examples of rings (noncommutative) in which prime ideals are completely

prime.

Example 5. Let R=

�

Z Z

0 Z

�

. Then P1 =

�

Z Z

0 0

�

, P2 =

�

0 Z

0 Z

�

and P3 =

�

0 Z

0 0

�

are prime ideals of R and all these are completely prime also.

A necessary and sufficient condition for a Noetherian ring R to be a σ(∗)-ring (where σ is

an automorphism of R) has been given in Theorem 2.4 of [3]:

Theorem 1. Let R be a Noetherian ring. Let σ be an automorphism of R. Then R is a σ(∗)-ring

if and only if for each minimal prime U of R, σ(U) = U and U is completely prime ideal of R.

Proof. See Theorem 2.4 of [3]. Please note that in Proposition 2.2 of [3] R should be

Noetharian. Proposition 2.2 of [3] has been used to prove Theorem 2.4 of [3].

Proposition 2. Let R be a Noetherian ring. Let σ be an automorphism of R such that R is a

σ(∗)-ring. Then U ∈ Min.Spec(R) implies that US(R) = U[x ;σ] is a completely prime ideal of

S(R) = R[x ;σ].

Proof. Proposition 1 implies that P(R) is completely semiprime ideal of R. Let U ∈ Min.Spec(R).

Then Theorem 1 implies that σ(U) = U and U is completely prime. Now we note that σ can

be extended to an automorphism σ of R/U . Now it is well known that S/US ≃ (R/U)[x ;σ]

and hence US is a completely prime ideal of S.

3. Skew Polynomial Rings over Weak σ-rigid Rings

Definition 3 (Ouyang [14]). Let R be a ring. Then R is said to be a weak σ-rigid ring if

aσ(a) ∈ N(R) if and only if a ∈ N(R) for a ∈ R.



N. Kumari, S. Gosani, V. Bhat / Eur. J. Pure Appl. Math, 6 (2013), 59-65 63

Example 6 (Example 2.1 of Ouyang [14]). Let σ be an endomorphism of a ring R such that R

is a σ-rigid ring. Let

A=
n







a b c

0 a d

0 0 a






| a, b, c, d ∈ R
o

be a subring of T3(R), the ring of upper triangular matrices over R. Now σ can be extended to an

endomorphism σ of A by σ((ai j)) = (σ(ai j)). Then it can be seen that A is a weak σ-rigid ring.

We now give a relation between a σ(∗)-ring and a weak σ-rigid ring in the following

Theorem:

Theorem 2. Let R be a Noetherian ring. Let σ be an endomorphism of R such that R is a σ(∗)-
ring. Then R is a weak σ-rigid ring. Conversely a 2-primal weak σ-rigid ring is a σ(∗)-ring.

Proof. Let σ be an endomorphism of R such that R is a σ(∗)-ring. Now R is completely

semiprime by Proposition 1. Therefore, R is 2-primal, i.e. N(R) = P(R). Thus aσ(a) ∈ N(R) =

P(R) implies that a ∈ P(R) = N(R). Hence R is weak σ-rigid ring.

Conversely let R be 2-primal weak σ-rigid ring. Then N(R) = P(R) and aσ(a) ∈ N(R)

implies that a ∈ N(R). Hence R is a σ(∗)-ring.

Corollary 1. Let R be a Noetherian ring. Let σ be an automorphism of R. Then R is a 2-primal

weak σ-rigid ring if and only if for each minimal prime U of R, σ(U) = U and U is completely

prime ideal of R.

Proof. Combine Theorem 1 and Theorem 2.

Let R be a Noetherian ring and σ an automorphism of R. We now give a characterization

for R to be a weak σ-rigid ring (an analog of Proposition 1 for weak σ-rigid rings).

Proposition 3. Let R be a Noetherian ring. Let σ be an automorphism of R. Then R is a weak

σ-rigid ring implies that N(R) is completely semiprime.

Proof. First of all we show that σ(N(R)) = N(R). We have σ(N(R)) ⊆ N(R) as σ(N(R))

is a nilpotent ideal of R. Now for any n ∈ N(R), there exists a ∈ R such that n = σ(a). So

I = σ−1(N(R)) = {a ∈ R such that σ(a) = n ∈ N(R)} is an ideal of R. Now I is nilpotent,

therefore I ⊆ N(R), which implies that N(R)⊆ σ(N(R)). Hence σ(N(R)) = N(R).

Now let R be a weak σ-rigid ring. We will show that N(R) is completely semiprime. Let

a ∈ R be such that a2 ∈ N(R). Then

aσ(a)σ(aσ(a)) = aσ(a)σ(a)σ2(a) ∈ σ(N(R)) = N(R).

Therefore aσ(a) ∈ N(R) and hence a ∈ N(R). So N(R) is completely semiprime.

Converse of the above Proposition need not be true (Example 3).

As mentioned earlier, we note that if σ is an endomorphism of a ring R, then it can be

extended to an endomorphism σ of S(R) = R[x ;σ] by σ(
∑m

i=0 x iai) =
∑m

i=0 x iσ(ai). We now

prove the following:



REFERENCES 64

Theorem 3. Let R be a Noetherian ring. Let σ be an automorphism of R. Then R is a weak

σ-rigid ring if and only if S(R) = R[x ;σ] is a weak σ-rigid ring.

Proof. First of all we note that Proposition 2.2 of Bhat [4] implies that S(N(R)) = N(S(R)).

Now let R be a weak σ-rigid ring. We show that R[x ;σ] is a weak σ-rigid ring.

Let f ∈ S(R) (say f =
∑m

i=0 x iai) be such that f σ( f ) ∈ N(S(R)). We use induction on

m to prove the Theorem. For m = 1, f = xa1 + a0. Now f σ( f ) ∈ N(S(R)) implies that

(xa1+ a0)(xσ(a1) +σ(a0)) ∈ N(S(R)) = S(N(R)), i.e.

x2σ2(a1) + xσ(a0)σ(a1) + xa1σ(a0) + a0σ(a0) ∈ S(N(R)) (1)

Therefore, σ2(a1) ∈ N(R). Now σ(N(R)) = N(R) implies that a1 ∈ N(R). So (1) implies

that a0σ(a0) ∈ N(R) implies that a0 ∈ N(R). Therefore, f ∈ S(N(R)) = N(S(R)).

Suppose the result is true for m = k. We prove for m = k + 1. Now f σ( f ) ∈ N(S(R))

implies that (x k+1ak+1+ . . .+ a0)(x
k+1σ(ak+1) + ldots+σ(a0)) ∈ N(S(R)) = S(N(R)), i.e.

x2k+2σk+2(ak+1) + x2k+1(σk(ak+1)σ(ak) +σ
k+1(ak)σ(ak+1)) + gσ(g) ∈ S(N(R)),

where g =
∑k

i=0 x iai. Therefore, σk+2(ak+1) ∈ N(R) implies that ak+1 ∈ N(R). Alsoσk(ak+1)σ(ak)+

σk+1(ak)σ(ak+1) ∈ N(R) implies that gσ(g) ∈ N(S(R)), but the degree of g is k, therefore,

by induction hypothesis, the result is true for all m.

Converse is obvious.

References

[1] S. Annin. Associated primes over skew polynomial rings, Communications in Algebra,

Vol. 30(5), 2511-2528. 2002.

[2] V. K. Bhat. Associated prime ideals of skew polynomial rings, Beitrage Algebra Geom-

etry, Vol. 49(1), 277-283. 2008.

[3] V. K. Bhat and N. Kumari. Transparency of σ(∗)-Rings and Their Extensions, Interna-

tional Journal of Algebra, Vol. 2(19), 919 - 924. 2008.

[4] V. K. Bhat and R. Raina. Ore Extensions over 2-primal Rings, Vietnam Journal of

Mathematics, Vol. 36(4), 455-461. 2008.

[5] W. D. Blair and L.W. Small. Embedding differential and skew polynomial rings into

artinian rings, Proceedings of the American Mathematical Society, Vol. 109 (4), 881-

886. 1990.

[6] C. Faith. Associated primes in commutative polynomial rings, Communications in Al-

gebra, Vol. 28, 3983-3986. 2000.

[7] K. R. Goodearl and R.B. Warfield. An introduction to Non-commutative Noetherian

rings, Camb. Uni. Press, 1989.



REFERENCES 65

[8] K. R. Goodearl and E. S. Letzter. Prime ideals in skew and q-skew polynomial rings,

Memoirs of the American Mathematical Society, 521, 1994.

[9] J. Krempa. Some examples of reduced rings, Algebra Colloquium, Vol. 3(4), 289-300.

1996.

[10] T. K. Kwak. Prime radicals of skew-polynomial rings, International Journal of Mathe-

matical Sciences, Vol. 2(2), 219-227. 2003.

[11] A. Leroy and J. Matczuk. On induced modules over Ore extensions, Communications

in Algebra, Vol. 32(7), 2743-2766. 2004.

[12] J. C. McConnell and J. C. Robson. Noncommutative Noetherian Rings, Wiley (1987);

revised edition: American Mathematical Society (2001).

[13] H. E. Nordstorm. Associated primes over Ore extensions, Journal of Algebra, Vol.

286(1), 69-75. 2005.

[14] L. Ouyang. Extensions of generalized α-rigid rings, International Electronic Journal

of Algebra, Vol. 3, 103-116. 2008.