Issue 2, Volume 5, 2011 82 Derivations and Iterated Skew Polynomial Rings Michael Gr. Voskoglou Abstract- Two are the objectives of the present set D of derivations of R, and that is why our concern is paper. First we study properties of a differentially turned to commutative rings only. Some characteristic simple commutative ring R with respect to a set D of examples of commutative Dsimple rings, where D is a derivations of R. Among the others we investigate the non singleton set of derivations, are given below. relation between the Dsimplicity of R and that of the local ring RP with respect to a prime ideal P of R and Example 1.1: The polynomial ring R=k[x ,x ,…,x ] over we prove a criterion about the D simplicity of R in 1 2 n a field k is a D simple ring, where case where R is a 1dimensional (Krull dimension) finitely generated algebra over a field of characteristic ∂ ∂ ∂ D={ ,...., }. zero and D is a singleton set. The above criterion was ∂ ∂ ∂xxx quoted without proof in an earlier paper of the author. 21 n Second we construct a special class of iterated skew Proof: If I is a non zero D-ideal of R and f is in I, we can polynomial rings defined with respect to finite sets of k i derivations of a ring R (not necessarily commutative) write f= ∑ i xf 1 , with fi in k[x2,…,xn] for each i. Then commuting to each other. The important thing in this i=0 class is that, if R is a commutative ring, then its k differential simplicity is the necessary and sufficient ∂ f =k!fk is a non zero element of condition for the simplicity of the corresponding skew ∂x k polynomial ring. 1 I ∩ k[x2,…,xn]. Repeating the same argument for fk and keep going in the same way one finds, after n at most Key-Words- Derivations, Differentially simple rings, Finitely- generated algebras, Iterated skew polynomial rings, Simple steps, that I ∩ k ≠ {0}. Thus I=R.- rings. Example 1.2: The power series ring R=k[[x1,x2,…,xn]] I. DIFFERENTIALLY SIMPLE COMMUTATIVE over a field k is a D-simple ring, where D is as in RINGS example 1.1 . The proof is the same as above.- zyxIR ],,[ ALL the rings considered in the first two sections of Example 1.3: Let R= be the 222 this paper are commutative with identity and all the fields ( zyx −++ )1 are of characteristic zero, unless it is otherwise stated. A coordinate ring of the real sphere (IR denotes the field of local ring is understood to be a Noetherian ring with a the real numbers). Consider the IR-derivations d 1 and d 2 unique maximal ideal; if R is not Noetherian, then we call → it a quasi-local ring. A Noetherian ring with finitely of the polynomial ring IR[x,y,z] defined by d 1 : x y+z, many maximal ideals is called a semi-local ring. For y → z-x, z → -x-y, and d 2 : x → y+2z, y → xyz-x, z → - special facts on commutative algebra we refer freely to 2 ++ 222 [2],[13], and [21], while for the concepts of Algebraic xy -2x. Then, since d i ( zyx -1)=0, for i=1,2 , Geometry used in the paper we refer freely to [4]. d i induces an IR-derivation of R, denoted also by d i . Let R be a ring (not necessarily commutative) and let D be a set of derivations of R. Then an ideal I of R is called Set D = {d 1 , d 2 }, then R is a Dsimple ring (cf. [38], a Dideal, if d(I) ⊆ I for all derivations d in D , and R is Lemma 3.1).- called a Dsimple ring if it has no non trivial D-ideals. The following example concerns a non trivial case of a When D is a singleton set, say D={d} , then, in order to differentially simple ring of prime characteristic: simplify our notation, I is called a d-ideal and R is called a d-simple ring respectively. In general R is called a Example 1.4: Let k be a field of prime characteristic, say differentially simple ring, if there exists at least one set D p, and let R and D be as in example 1.1. Then p p p of derivations of R, such that R is a D-simple ring. I=(x1 ,x2 ,…,xn ) is obviously a D-ideal of R and Every D-simple ring R contains the field ∂ therefore each , i=1,2,…,n , induces a derivation, say F= C(R) [∩∩ ker d], where C(R) denotes the center ∈Dd ∂xi of R, and therefore, if R is of characteristic zero, then it contains the field of rational numbers. Non commutative differentially simple rings exist in abundance, e.g. every simple ring R is D-simple for any INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS AND INFORMATICS Issue 2, Volume 5, 2011 83 ∂f (i.e. a localization of a finitely generated k-algebra at a di , of the factor ring R/I by di(f+I)= +I, for all f in R. prime ideal of R), Hart [7] constructed a derivation d of ∂ xi R, such that R is a d-simple ring. The above results show * * Set D ={d1,d2,…,dn}, then R/I is a D -simple ring. that for a wide class of rings of characteristic zero the d- Proof: If A is a non zero D*-ideal of R/I, then A΄= simplicity is connected with the regularity, a property {f∈k[x1,x2,…,xn] : f+I∈A} is obviously a Dideal of which requires from a ring R to have a “special” kind of k[x1,x2,…,xn] containing properly I. It becomes evident dimension. In fact, in this case every maximal ideal M of that there exists f in A΄ with all its terms of the R can be generated by dim RM elements, where RM mm m form 21 ....xxcx n , with c in k and 0 <≤ pm , for denotes the localization of R at M. In particular, if R is a 21 n i finitely generated algebra, then M can be generated by each i=1, 2,…,n. Let m be the greatest integer which dim R elements. appears as exponent of x1 in the terms of f. Without loss Consider now the localization RP of a ring R at a proper of the generality we may assume that m ≠ 0. Then prime ideal P of R. Then every derivation d of R extends m ∂ f to a derivation of RP by the usual rule of differentials for ≠ 0 m = m!f1(x2,…,xn) is in A΄. If f1 is not in k, we rational functions. We prove the following result: ∂x 1 repeat the same argument for f1 and we keep going in the Theorem 1.6: Let R and RP be as above. Then, if R is a same way until we find, after a finite number of steps, Dsimple ring with respect to a set D of derivations of R, that A΄ ∩ k ≠ }0{ . Thus A = R/I and the result follows.- RP is also Dsimple. Conversely, if RP is D-simple with There is no general criterion known to decide whether or respect to a set D of derivations of RP whose restrictions not a given ring R is differentially simple. However it to R are derivations of R, then R is also a D-simple ring. seems that there is a connection between the differential Proof: Assume first that R is a Dsimple ring, while Rp is simplicity and the Krull dimension of R (denoted by dim not. Let q be a prime ideal of RP, such that d(q) ⊆ q for all R). The following result demonstrates this connection, d in D. Then there exists a prime ideal Q of R not meeting when R is of prime characteristic: P, such that q = Qe r Theorem 1.5: Let R be a ring of prime characteristic, say = { : r∈Q, s∈R-P} (cf. [2], Proposition 3.11). Since R p, and let D be a set of derivations of R, such that R is D- s simple. Then R is a 0-dimensional, quasi-local ring. is a Dsimple ring, we can find r in Q, such that d(r)∉Q Proof: Let M be a maximal ideal of R and let I be the r p for all d in D. But d( )=d(r) is in ideal of R generated by the set {m : m∈M}. Then, since 1 R is of characteristic p, I is a proper Dideal of R, Qe =q, therefore d(r) is in Q, which is absurd. therefore the D-simplicity of R implies that I=(0). Thus M For the converse, assume that there exists a non zero is contained in the nil radical, say N, of R and therefore prime ideal, say Q, of R, such that d(Q) ⊆ Q, for all M=N. Let now P be a prime ideal of R contained in M. ∈ ⇔ Then, since N is equal to the intersection of all prime derivations d of D. Then, for all r in Q, d(r) Q ideals of R and M=N, we get that M=P. Thus N is the sd(r)∈Q, for all s in P-R unique prime ideal of R and this proves the theorem.- − srdrsd )()( r e ⇔ d )( ∈= Q .