S e ° MR
СИБИРСКИЕ ЭЛЕКТРОННЫЕ МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ
Siberian Electronic Mathematical Reports
http://semr.math.nsc.ru
Том 6, стр. 505–509 (2009) УДК 512.55
MSC 16N40, 16P40, 16S36
A NOTE ON σ(∗)-RINGS AND THEIR EXTENSIONS
V. K. BHAT, NEETU KUMARI
Abstract. LetRbe an associative ring with identity16= 0, andσan endomorphism of R. We recall σ(∗) property on R (i.e. aσ(a) ∈ P(R) impliesa∈P(R)fora∈R, whereP(R)is the prime radical ofR). Also recall that a ringRis said to be 2-primal if and only if the prime radical P(R)and nil radical are same, i.e. if the prime radical is a completely semiprime ideal. It can be seen that aσ(∗)ring is a 2-primal ring.
Let R be a ring and σ an automorphism ofR. Then we know that σ can be extended to an automorphism of the skew polynomial ring R[x;σ]. In this paper we show that ifR is a Noetherian ring andσ is an automorphism ofRsuch thatRis aσ(∗)-ring, thenR[x;σ]is also a σ(∗)-ring.
Keywords:minimal prime, prime radical, automorphism,σ(∗)-ring.
1. Introduction
A ringRalways means an associative ring with identity16= 0. The set of prime ideals ofRis denoted bySpec(R). The sets of minimal prime ideals ofRis denoted byM in.Spec(R). The prime radical and the nil radical ofR are denoted byP(R) and N(R) respectively. The set of positive integers is denoted by N, the ring of integers is denoted byZ, the field of rational numbers is denoted byQand the field of complex numbers is denoted byC. LetRbe a ring andσan automorphism ofR.
LetI be an ideal ofRsuch thatσm(I) =I for somem∈N. We denote ∩mi=1σi(I) byI0.
V. K. Bhat, Neetu Kumari, A note onσ(∗)-rings and their extensions.
c
°2009 V. K. Bhat, Neetu Kumari.
The authors would like to express their sincere thanks to the referee for the suggestions.
Received August, 6, 2008, published November, 27, 2009.
505
This article concerns the study of skew polynomial ring (Ore extension) over a σ(∗)-ringR, where σis an automorphism ofR.
Definition 1.1. (Krempa [3]) LetRbe a ring andσan endomorphism ofR. Then σ is said to be a rigid endomorphism ifaσ(a) = 0implies that a= 0, for a∈R.
The ringR is said to be aσ-rigid ring if there exists aσ-rigid endomorphism R.
Example 1.2. LetR=C, andσ:C→Cbe the map defined byσ(a+ib) =a−ib, a,b∈R. Then it can be seen thatσ is a rigid endomorphism ofR.
Definition 1.3. (Kwak [4]) Let Rbe a ring and σan endomorphism of R. Then R is said to be aσ(∗)-ring ifaσ(a)∈P(R)impliesa∈P(R)fora∈R.
Example 1.4. Let R=
µ F F
0 F
¶
, where F is a field. Then P(R) =
µ 0 F 0 0
¶
Letσ:R→Rbe defined byσ
³ µ a b 0 c
¶ ´
=
µ a 0 0 c
¶
.Then it can be seen that σis an endomorphism of R and R is a σ(∗)-ring.
We note that the above ring is notσ-rigid. For let06=a∈F. Then µ 0 a
0 0
¶ σ
µ 0 a 0 0
¶
=
µ 0 0 0 0
¶ , but
µ 0 a 0 0
¶ 6=
µ 0 0 0 0
¶ .
Recall that a ring R is 2-primal if and only if N(R) = P(R) if and only if the prime radical is a completely semiprime ideal. An idealI of a ring Ris called completely semiprime ifa2∈Iimpliesa∈Ifora∈R. We note that a commutative ring is 2-primal and so is a reduced ring.
Recall thatR[x;σ]is the usual polynomial ring with coefficients in R, in which multiplication is subject to the relation ax = xσ(a) for all a ∈ R. We take any f(x)∈R[x;σ] to be of the formf(x) =Pn
i=0xiai. We denoteR[x;σ]byS(R). If an idealI of a ringRisσ-stable (i.e.σ(I) =I), then we denote as usualI[x;σ]by S(I).
We also note that ifσ is an automorphism ofR, then it can be extended to an automorphism of R[x;σ]such that σ(x) =x; i.e.σ(Σni=0xiai) = Σni=0xiσ(ai). The study of skew polynomial rings has been of interest to many authors. For example [1, 2, 4].
Definition 1.5. (Goodearl and Warfield [2]) A ring R is said to be right(left) Noetherian ring if it satisfies the ascending chain condition on right(left) ideals.
Explicitly this means: given an increasing sequence of right(left) ideals I1⊆I2⊆I3⊆...
there exists ann∈N for which
In =In+1=In+2=...
Equivalently R is right(left) Noetherian if every right(left) ideal of R is finitely generated.R is said to be Noetherian if it is both right and left Noetherian.
Some examples:The ring of integersZ, any field, any principal ideal domain, the ring of polynomials in finitely-many variables over the integers or a field.
The ring of polynomials in infinitely-many variables,x1, x2, x3, ...is not Noetherian as the ascending sequence of ideals (x1) ⊆ (x1, x2) ⊆ (x1, x2, x3) ⊆ ... does not terminate.
We note that a right Noetherian ring need not be left Noetherian. For example considerR={
µ a b 0 c
¶
witha∈Zandb, c∈Q}. ThenRis right Noetherian but not left Noetherian.
We now state the main result of this paper in the form of the following Theorem [Theorem (2.6]:
Theorem:If R is a Noetherian ring and σis an automorphism of R such that R is aσ(∗)-ring, then R[x;σ]is also a σ(∗)-ring.
2. Skew polynomial rings over σ(∗)-rings We begin this section with the following Proposition:
Proposition 2.1. Let R be a ring and σ an automorphism of R. Then R is a σ(∗)-ring implies R is 2-primal.
Proof.Leta∈Rbe such thata2∈P(R). Thenaσ(a)σ(aσ(a)) =aσ(a)σ(a)σ2(a)∈ σ(P(R)) =P(R).Thereforeaσ(a)∈P(R)and hencea∈P(R).
The following example shows that there exists an endomorphismσ of a ring R such that the converse of the above Proposition does not hold.
Example 2.2. Let R = F[x], F a field. Then R is a commutative domain, and therefore is 2-primal with P(R) = 0. Letσ:R→R be defined by σ(f(x)) =f(0).
Let f(x) =xa,06=a∈F. Thenf(x)σ(f(x))∈P(R), butf(x)∈/ P(R). Therefore R is not a σ(∗)-ring.
Recall that an idealJ ofR is called completely prime ifab∈J impliesa∈J or b∈J fora, b∈R.
We know that for any ringR, the prime radicalP(R)is the intersection of prime ideals ofR. Now Proposition (3.3) of Goodearl and Warfield [2] implies that every prime ideal ofRcontains a minimal prime ideal. Therefore, the prime radicalP(R) is the intersection of minimal prime ideals ofR(Proposition (3.10) of Goodearl and Warfield [2]). With this, we have the following:
Theorem 2.3. Let R be a Noetherian ring, and σ an automorphism of R. Then R is aσ(∗)-ring if and only if for each minimal prime U of R, σ(U) =U and U is completely prime ideal of R.
Proof. Let R be a Noetherian ring such that for each minimal prime U of R, σ(U) =U and U is completely prime ideal of R. Let a∈R be such that aσ(a)∈ P(R) =∩ni=1Ui, whereUi are the minimal primes of R. Now for eachi,a∈Ui or σ(a)∈Ui asUi are completely prime. Nowσ(a)∈Ui =σ(Ui)implies thata∈Ui. Thereforea∈P(R). HenceR is aσ(∗)-ring.
Conversely, suppose thatRis aσ(∗)-ring and letU =U1be a minimal prime ideal of R. Now by Proposition (2.1),P(R)is completely semiprime. NowM in.Spec(R) is finite by Theorem (3.4) of Goodearl and Warfield [2]. Let U2, U3, ..., Un be the
other minimal primes ofR. Suppose thatσ(U)6=U. Thenσ(U)is also a minimal prime ideal ofR. Renumber so thatσ(U) =Un. Leta∈ ∩n−1i=1Ui. Then σ(a)∈Un, and soaσ(a)∈ ∩ni=1Ui=P(R). Thereforea∈P(R), and thus∩n−1i=1Ui⊆Un, which implies thatUi⊆Un for somei6=n, which is impossible. Henceσ(U) =U.
Now suppose thatU =U1is not completely prime. Then there exista, b∈R\U withab∈U. Letcbe any element ofb(U2∩U3∩...∩Un)a. Thenc2∈ ∩ni=1Ui=P(R).
Soc∈P(R)and, thusb(U2∩U3∩...∩Un)a⊆U. ThereforebR(U2∩U3∩...∩Un)Ra⊆ U and, as U is prime, a /∈U, Ui ⊆U for somei6= 1 or b∈U. None of these can occur, soU is completely prime.
Note that in above Theorem the condition of completely primeness of minimal prime ideals can not be deleted. Towards this we have the following:
Remark 2.4. LetR be a Noetherian ring and σ an automorphism ofR such that σ(U) =U for each minimal prime idealU of R. Then R[x;σ]need not be a σ(∗)- ring. (Example (2.2))
We note that ifRis a Noetherian ring, then as mentioned above,M in.Spec(R) is finite. Now if σ is an automorphism of R, then σj(U) ∈ M in.Spec(R) for any U ∈ M in.Spec(R) for all j ∈ N. Therefore, there exists some m ∈ N such that σm(U) =U for allU ∈M in.Spec(R). We denote∩mi=1σi(U)byU0. We now have the following:
Theorem 2.5. Let R be a Noetherian ring and σ an automorphism of R. Then : P ∈ M in.Spec(S(R)) if and only if there exists Q ∈ M in.Spec(R) Such that S(P∩R) =P andP∩R=Q0.
Proof.See Theorem (2.4) of Bhat [1].
Theorem 2.6. Let R be a Noetherian ring andσan automorphism of R such that R is aσ(∗)-ring. Then R[x;σ]is also a σ(∗)-ring.
Proof. First of all we show that σ(P) = P, for all P ∈ M in.Spec(S(R)). Let P ∈M in.Spec(S(R)). Then by Theorem (2.5) there existsU ∈M in.Spec(R)such thatP =U0[x;σ]. Now Ris aσ(∗)-ring implies thatσ(U) =U by Theorem (2.3), and thereforeU0=U. SoP =U[x;σ]and thusσ(P) =P.
We now show thatP is completely prime. Let
f(x) =xnan+xn−1an−1+...+a0, and
g(x) =xmbm+xm−1bm−1+...+b0 inR[x;σ]be such that f(x)g(x)∈P =U[x;σ], andg(x)∈/U[x;σ].
This implies that
xn+mσm(an)bm+xn+m−1σm(an−1)bm+xn+m−1σm−1(an)bm−1+...+a0b0∈U[x;σ].
Now g(x) ∈/ U[x;σ] (saybm ∈/ U). Now σm(an)bm ∈ U. Also U is completely prime by Theorem(2.3), therefore,σm(an)∈U; i.e.an ∈U.
Now σm(an−1)bm+σm−1(an)bm−1 ∈ U implies that σm(an−1)bm ∈ U. Now bm∈/U implies thatσm(an−1)∈U; i.e. an−1∈U.
With the same process in a finite number of steps it can be seen thatai∈U for all i,0≤i≤n−2also.
Thereforeai∈U for all i,0≤i≤n; i.e.f(x)∈P=U[x;σ].
Thus σ(P) = P and P is completely prime for all P ∈ M in.Spec(S(R)).
Moreover S(R) is Noetherian by Theorem (1.14) of Goodearl and Warfield [2].
Hence by Theorem (2.3), we get thatR[x;σ]is also a σ(∗)-ring.
Remark 2.7. (1) LetRbe a Noetherian ring andσan automorphism ofRsuch thatRis aσ(∗)-ring. ThenR[x;σ]is also aσ(∗)-ring. Therefore Proposition (2.1) implies thatR[x;σ]is 2-primal.
(2) If R is 2-primal Noetherian ring, then R[x;σ] need not be 2-primal. For example considerZ2and letR=Z2⊕Z2. ThenRis a commutative reduced ring with P(R) = 0, and therefore, R is 2-primal. Defineσ : R → R by σ(a, b) = (b, a). Then it can be seen that P(R[x;σ]) = 0, but P(R[x;σ]) is not completely semiprime as ((1,0)x)2 = 0 =P(R[x;σ]), but (1,0)x /∈ P(R[x;σ]). ThusR[x;σ]is not 2-primal.
References
[1] V. K. Bhat,Associated prime ideals of skew polynomial rings. Beitrдge Algebra Geom.49: 1 (2008), 277–283. MR2410584 (2009e:16046).
[2] K. R. Goodearl and R. B. Warfield Jr.,An introduction to non-commutative Noetherian rings. Second Edition, Cambridge Uni. Press, 2004. MR2080008 (2005b:16001).
[3] J. Krempa, Some examples of reduced rings. Algebra Colloq. 3: 4 (1996), 289–300.
MR1422968 (98e:16027).
[4] T. K. Kwak,Prime radicals of skew-polynomial rings. Int. J. Math. Sci.2: 2 (2003), 219–227.
MR2061508. (2006a:16035).
V. K. Bhat and Neetu Kumari School of Mathematics,
SMVD University, P/o SMVD University, 182320, Katra, J and K, India
E-mail address:[email protected]
