Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 22, 2006, 3–6

\begin{matrix}c \\ ◯ \end{matrix} V. K. Bhat

V. K. Bhat

A NOTE ON KRULL DIMENSION OF SKEW POLYNOMIAL RINGS

(submitted by M. M. Arslanov)

Abstract. Let A be a commutative Noetherian ring such that Krull dimension of A is α. Let M be a finitely generated critical module over A[x, σ], (where σ is an automorphism of A) and Krull dimension of M is α + 1. Then M has a prime annihilator.

1. Introduction

All rings are with identity, and all modules unitary. If a module M over a ring R has a Krull dimension α, we denote it by | M |= α. Classical Krull dimension of a ring R is denoted by cl.K(R). For a module M over a ring R with | M |= α, we say M is α-homogeneous if M contains no non-zero submodule of Krull dimension less than α. For more details and some concerning results on Krull dimension, the reader is referred to [3]. Let B be a right Noetherian ring, T_{N}(B) denotes the torsion submodule of B at the prime radical N(B) of B. We use a similar notation for T_{N}(B[x, σ]), which is the torsion submodule of B[x, σ], where σ is an automorphism of B and B[x, σ] is the usual skew polynomial ring of B in which coefficients of polynomials are taken on the right, and therefore B[x, σ] = {∑ x^{i}a_{i}, a_{i} ∈ B, 0 ≤ i ≤ n, for some positive integer n},

16P40, 16P50, 16U20. prime annihilator.

2000 Mathematical Subject Classification. Primary 16-XX; Secondary

Key words and phrases. Automorphism, Krull dimension, critical module,

subject to the relation ax = xσ(a). C(N(B)) denotes the set of elements of B regular modulo N(B).

Let now A be a commutative Noetherian ring with | A |= α. Then | A[x, σ] |= α + 1, where σ is an automorphism of A. We show that any finitely generated critical module M over A[x, σ] with | M |= α + 1 has a prime annihilator.

2. Critical modules over R[x, σ]

We begin this section with the following Proposition:

Proposition 2.1. Let B be a right Noetherian ring and σ be an automorphism of B. Then σ(T_{N}(B)) = T_{N}(B).

Proof. Using the fact that σ(B) = B and if c ∈ C(N), then σ(c) ∈ C(N), it can be easily proved that σ(T_{N}(B)) = T_{N}(B). □

Proposition 2.2. Let B be a right Noetherian ring and σ be an automorphism of B. If N is the prime radical of B, then σ(N) = N.

Proof. σ(N) ⊆ N because σ(N) is a nilpotent ideal of B. Let n ∈ N. Then n = σ(a) for some a ∈ B. Therefore σ^{−1}(n) = a, and σ^{−1}(N) = {a ∈ B, such that a = σ^{−1}(n), some n ∈ N} = I (say) is an ideal of B. Now σ(I) ⊆ σ(N). Hence σ(N) = N. □

Proposition 2.3. Let B be a semiprime Noetherian ring. Let f ∈ B[x, σ] be regular in B[x, σ]. Then there exists g ∈ B[x, σ] such that gf has leading coefficient regular in B.

Proof. Note that ax = xσ(a) for all a ∈ B. Let S = {x^{m}a_{m} + ... + a_{0} ∈ B[x, σ]f, some m} ∪ {0}. Let a_{m} ∈ S and c ∈ R. Then we have some g = ∑ x^{i}a_{i} ∈ B[x, σ], (i = 1, 2, ..., n) and f = ∑ x^{j}d_{j}, (j = 1, 2, ..., t) regular in B[x, σ] such that leading coefficient of gf is a_{m}; i.e. σ^{t}(a_{n})d_{t} = a_{m}. Now h = ∑ x^{j}σ^{−t}(c)a_{j} ∈ B[x, σ], (j = 1, 2, ..., t), and therefore hf has leading coefficient in S; i.e. cσ^{t}(a_{n})d_{t} ∈ S. Thus S is a left ideal. We now show that S is essential. Let 0 ≠ I ⊆ B be a left ideal of B. Then it is easy to see that I[x, σ] is a left ideal of B[x, σ]. Now f ∈ B[x, σ] is regular, therefore B[x, σ]f is an essential left ideal of B[x, σ]; i.e. I[x, σ] ∩ B[x, σ]f ≠ (0). Let ∑ x^{i}a_{i} ∈ I[x, σ] ∩ B[x, σ]f, (i = 1, 2, ..., k). Then a_{k} ∈ I and a_{k} ∈ S, and therefore S is essential as a left ideal. So S contains a left regular element by Goldie‘s Theorem, see for example [1, Theorem (1.37)]. Now B is semiprime Noetherian implies that S contains a regular element. Hence there exists g ∈ B[x, σ] such that gf has leading coefficient regular in B. □

Corollary 2.4. Let B be a right Noetherian ring and f ∈ B[x, σ] regular modulo N^{∗}, where N^{∗} is the prime radical of B[x, σ]. Then there exists g ∈ B[x, σ] such that gf has leading coefficient regular in B/N, where N is the prime radical of B.

Proposition 2.5. Let B be a right Noetherian ring, and σ an automorphism of B. Then (T_{N}(B))[x, σ] is a right ideal of B[x, σ], and (T_{N}(B))[x, σ] = (T_{N}(B[x, σ]).

Proof. By 2.1 above σ(T_{N}(B)) = T_{N}(B), therefore (T_{N}(B))[x, σ] is a right ideal of B[x, σ]. Now on the same lines as in [6, Proposition (1.1)] with some manipulations on σ, it can be easily proved that T_{N}(B[x, σ]) ⊆ (T_{N}(B))[x, σ]. □

Theorem 2.6. Let A be commutative Noetherian ring with | A |= α. If M is a finitely generated critical right module over A[x, σ] with | M |= α + 1. Then M has a prime annihilator, where σ is an automorphism of A.

Proof. Let F_{α} = Sum of all submodules of A of Krull dimension less than α. Then as in [6, Theorem(1.2)], F_{α} = ∩A_{α}, where A_{α} are ideals of A such that A/A_{α} is α-homogeneous. Also observe that A/σ(F_{α}) is isomorphic to A/F_{α}. Therefore A/σ(F_{α}) is α-homogeneous. So σ(F_{α}) ⊆ F_{α}. Now let F_{α}^{∗} = Sum of submodules of A[x, σ] having Krull dimension less than α + 1. Then F_{α}[x, σ] ⊆ F_{α}^{∗} and F_{α}^{∗} = ∩A_{α}^{∗}, where A[x, σ]/A is α + 1homogeneous. Let Ann(M) = B, where Ann(M) denotes the annihilator of M. Then M is a critical A[x, σ]/B module, which is also faithful since | M |= α + 1, therefore | A[x, σ]/B |= α + 1. Now M is critical with | M |= α + 1, so by [2] A[x, σ]/B is isomorphic to a submodule of direct sum of n copies of M. This easily yields that A[x, σ]/B is α + 1-homogeneous. Hence F_{α}[x, σ] ⊆ B. Thus M is an A[x, σ]/F_{α}[x, σ] module. If N(A/F_{α}) is the prime radical of A/F_{α}, then since A/F_{α} is α-homogeneous, (A/F_{α})/N(A/F_{α}) is also α-homogeneous, because A is a commutative ring. Now using [6, Theorem (1.2)] and the fact that every critical module is compressible over A/F_{α} by [4, Theorem (2.5)], we get by [2, Proposition (3.6)] that (A/F_{α})/T_{N}(A/F_{α}) has an artinian quotient ring, therefore by [5, Theorem (3.1)] (A/F_{α})[x, σ]/(T_{N}(A/F_{α}))[x, σ] has an artinian quotient ring. Now 2.5 implies that

(A/F_{α})[x, σ]/T_{N}((A/F_{α})[x, σ]) = (A/F_{α})[x, σ]/(T_{N}(A/F_{α}))[x, σ].

Now since

(A/F_{α})[x, σ]/(N(A/F_{α}))[x, σ]

6 V. K. BHAT

is α + 1-homogeneous as (A/F_{α})/N(A/F_{α}) is α-homogeneous, so again by an application of [2, Proposition (3.6)], we get that M as a module over (A/F_{α})[x, σ] is compressible. Hence M has a prime annihilator. □

References

[1] A. W. Goldie, The structure of Noetherian rings, Lecture notes in Mathematics, 246 Springer Verlag, 1970-71.

[2] R. Gordon, Some aspects of non-commutative Noetherian rings, Springer Verlag, 1975.

[3] R. Gordon and J. C. Robson, Krull dimension, Memoirs Amer. Math. Soc. 133 (1974).

[4] A. V. Jategoankar, Skew polynomial rings over orders in artinian rings, J.Algebra 21, 51-59 (1972).

[5] A. V. Jategoankar, Jacobson,s conjecture and modules over FBN rings, J.Algebra 30, 105-121 (1974).

[6] C. L. Wangneo, Polynomial rings over FBN rings, Algebra and its applications, Marcel Dekker INC (1984)

V. K. BHAT, School of Applied Physics and Mathematics, SMVD University, P/o Kakryal, Udhampur, J and K, India- 180001 E-mail address: vijaykumarbhat2000@yahoo.com

Received May 20, 2006