STABLE RANGE IN FORMAL POWER SERIES WITH ANY NUMBER OF INDETERMINATES

Amir M. Rahimi

Abstract. Throughout this work (unless otherwise indicated), all rings are commutative rings with identity. Let R be a ring, Λ an index set with cardinality |Λ| and let {X_{λ}}_{λ∈Λ} be an arbitrary set of indeterminates over R. In this work for each fixed i = 1, 2 or 3 we show the ring T_{i} = R[[{X_{λ}}_{λ∈Λ}]]_{i} of formal power series with |Λ| indeterminates over R is n–stable (respectively, a B– ring), if and only if R is n–stable (respectively, a B–ring). For each s ≥ 1, a sequence (a_{1}, a_{2}, …, a_{s}, a_{s+1}) of elements in R is said to be stable whenever the ideal (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}) = (a_{1} + b_{1}a_{s+1}, ⋯, a_{s} + b_{s}a_{s+1}) for some b_{1}, b_{2}, ⋯, b_{s} ∈ R. A sequence (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}), a_{i} ∈ R, is said to be unimodular whenever ideal (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}) = R. For any fixed positive integer n we shall say n is in the stable range of R (or simply, R is n-stable), whenever for all s ≥ n any unimodular sequence (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}), a_{i} ∈ R, is stable. R is said to be a B–ring, if for any unimodular sequence (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}), s ≥ 2, a_{i} ∈ R with (a_{1}, a_{2}, ⋯, a_{s−1}) ⊄ Jacobson radical of R, there exists b ∈ R such that (a_{1}, a_{2}, ⋯, a_{s} + ba_{s+1}) = R.

1. Introduction. In this work (unless otherwise indicated), all rings are commutative rings with identity. Let R be a ring and Λ be an index set with cardinality |Λ|. Let Z_{0} denote the abelian monoid of non–negative rational integers. We assume that the reader is familiar with the concept of the ring R[X_{1}, X_{2}, ⋯, X_{n}] of polynomials with a finite number of indeterminates over R, and also with R[{X_{λ}}_{λ∈Λ}] the ring of polynomials with |Λ| indeterminates over R. For references on the ring of polynomials see [3, 6]. By the degree of a monomial aX_{1}^{i_{1}}X_{2}^{i_{2}} ⋯ X_{n}^{i_{n}}(a ∈ R, i_{1}, i_{2}, ⋯, i_{n} ∈ Z_{0}) we mean the sum of its exponents which is i_{1} + i_{2} + ⋯ + i_{n}. The degree of a nonzero polynomial f which is denoted by ∂f, is the maximum of the degrees of the monomials of which f is the sum. If all the monomials in the sum have the same degree, f is said to be a form. It is clear that if f and g are forms, then fg is either zero or a form of degree ∂fg = ∂f + ∂g. A polynomial f of degree m can be expressed uniquely as f = f_{0} + f_{1} + ⋯ + f_{m}, where each f_{i} is either zero or a form of degree i and f_{m} cannot be zero.

Now we define the ring of formal power series with a finite number of indeterminates. Let S = R[X_{1}, X_{2}, ⋯, X_{n}] and define S^{∗} to be the set {{f_{i}}_{0}^{∞}}, where for each i ∈ Z_{0}, f_{i} ∈ S is either zero or a form of degree i. For each {f_{i}}_{0}^{∞} and {g_{i}}_{0}^{∞} in S^{∗}, {f_{i}} = {g_{i}} if and only if f_{i} = g_{i} for all i ∈ Z_{0}, {f_{i}} + {g_{i}} = {f_{i} + g_{i}} and {f_{i}}{g_{i}} = {h_{i}} where

h_{i} = ∑_{j=0}^{i} f_{j}g_{i−j}

for each i ∈ Z_{0}. Under the above relation and operations S^{∗} is a ring and is denoted by R[[X_{1}, X_{2}, ⋯, X_{n}]]. S^{∗} is called the ring of formal power series with n indeterminates over R. In fact each {f_{i}}_{0}^{∞} can be identified with the formal power series

∑_{i=0}^{∞} f_{i}

where for each

∑_{i=0}^{∞} f_{i}

and

∑_{i=0}^{∞} g_{i},

addition, multiplication, and equality can be defined as above, correspondingly. It is not difficult to show that S^{∗} has an identity if and only if R has an identity. If {X_{λ}}_{λ∈Λ} is an arbitrary set of indeterminates over R, there are three ways of defining the ring of formal power series for the set {X_{λ}}_{λ∈Λ} over R. We denote these rings by T_{i} = R[[{X_{λ}}_{λ∈Λ}]]_{i} where i = 1, 2 or 3. We define T_{1} to be the set of all rings of the form R[[X_{λ_{1}}, X_{λ_{2}}, ⋯, X_{λ_{n}}]], where {λ_{1}, λ_{2}, ⋯, λ_{n}} runs over all finite subsets of Λ. T_{2} is the set of all formal sums

∑_{i=0}^{∞} f_{i}

where for each i, f_{i} ∈ R[{X_{λ}}_{λ∈Λ}] is zero or a form of degree i. Note that equality, addition and multiplication in T_{1} and T_{2} are defined in the same obvious way as above. For example, let

f = ∑ f_{i}

and

g = ∑ g_{i}

be elements in T_{1}, then f ∈ R[[X_{λ_{1}}, X_{λ_{2}}, ⋯, X_{λ_{n}}]] and g ∈ R[[X_{λ′_{1}}, X_{λ′_{2}}, ⋯, X_{λ′_{m}}]] for some subsets {λ_{1}, λ_{2}, …, λ_{n}} and {λ′_{1}, λ′_{2}, …, λ′_{m}} of Λ. In this case it is obvious that both f and g are in

R[[X_{λ_{1}}, X_{λ_{2}}, ⋯, X_{λ_{n}}, X_{λ′_{1}}, ⋯, X_{λ′_{m}}]].

Next we show that T_{1} (respectively, T_{2}) has an identity if and only if R has an identity. The necessary condition is immediate since R is a homomorphic image of T_{1} (respectively, T_{2}) under the mapping ∑ f_{i} ↦ f_{0}. Conversely, it is obvious that if 1 in R is the identity element, then

∑_{i=0}^{∞} f_{i}

with f_{0} = 1 and f_{i} = 0 for all i ≥ 1, is an identity element in T_{1} (respectively, T_{2}).

Let A be an abelian monoid such that for each a ∈ A there are only a finite number of ordered pairs (b, c) of elements in A with b + c = a. Let T be the set of all functions from A into R. For all a ∈ A and all elements f, g ∈ T define equality, addition, and multiplication as follows: f = g if and only if f(a) = g(a), (f+g)(a) = f(a) + g(a) and

(fg)(a) = ∑_{b+c=a} f(b)g(c).

Under this definition T is a ring and whenever A is a direct sum of n copies of Z_{0}, then T is isomorphic to R[[X_{1}, X_{2}, ⋯, X_{n}]]. Now we show that T has an identity if and only if R has an identity. Assume 1_{R} is the identity element of R, then the function 1_{T} : A → R, given by 1_{T} (0) = 1_{R} and zero otherwise, is the identity element in T. Conversely, assume f is the identity element in T. In this case we claim f(0) is the identity element in R. Let r be an arbitrary element in R and define f_{r} : A → R to be a function that maps zero to r and is zero at other points in A. Now we have

rf(0) = f_{r}(0)f(0) = ∑_{b+c=0} f_{r}(b)f(c) = (f_{r}f)(0) = f_{r}(0) = r.

Now we are ready to define T_{3}. In the above definition of the ring T, if we assume

A = ∑(Z_{0})_{λ}

is the weak direct sum of |Λ| copies of Z_{0}, then we get a ring which is denoted by T_{3} = R[[{X_{λ}}_{λ∈Λ}]]_{3}. Next we show that each element f ∈ T_{3} can be written as a formal sum

∑_{i=0}^{∞} f_{i}

where f_{i} is either zero or a form of degree i. Indeed, f_{i} in

∑_{i=0}^{∞} f_{i}

can be a form with infinitely many terms and this is the main difference between T_{2} and T_{3}. Let

{a_{λ}} ∈ ∑(Z_{0})_{λ}

where possibly a_{λ_{1}} ≠ 0, a_{λ_{2}} ≠ 0, ⋯, a_{λ_{n}} ≠ 0 for some λ_{1}, λ_{2}, ⋯, λ_{n} ∈ Λ. Now for each r ∈ R, define rX_{λ_{1}}^{a_{λ_{1}}} X_{λ_{2}}^{a_{λ_{2}}} ⋯ X_{λ_{n}}^{a_{λn}} to be the function from

∑(Z_{0})_{λ}

into R such that (rX_{λ_{1}}^{a_{λ_{1}}} X_{λ_{2}}^{a_{λ_{2}}} ⋯ X_{λ_{n}}^{a_{λn}})({a_{λ}}) = r and zero otherwise. Thus, it is clear that each f ∈ T_{3} can be expressed as a formal sum of monomials of the form rX_{λ_{1}}^{a_{λ_{1}}} X_{λ_{2}}^{a_{λ_{2}}} ⋯ X_{λ_{n}}^{a_{λn}}. From the definition of T_{1}, T_{2} and T_{3} it is not difficult to show that T_{1} can be embedded into T_{2} and T_{2} can be embedded into T_{3} or simply, we can write T_{1} ⊂ T_{2} ⊂ T_{3}. Actually whenever |Λ| is finite, T_{1} = T_{2} = T_{3}. In other words T_{i} is independent of the choice of i. For a general reference about T_{1}, T_{2}, and T_{3}, see [2].

In this paper, without having any confusion in the context, we use parentheses to show both the sequence (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}), s ≥ 1, of elements in R, and the ideal (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}) generated by a_{1}, a_{2}, ⋯, a_{s}, a_{s+1} ∈ R. A sequence (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}) of elements in R is said to be stable, whenever (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}) = (a_{1} + b_{1}a_{s+1}, ⋯, a_{s} + b_{s}a_{s+1}) for some b_{1}, b_{2}, ⋯, b_{s} in R. A sequence (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}), a_{i} ∈ R, is said to be unimodular, if (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}) = R. For a fixed integer n ≥ 1, we shall say n is in the stable range of R (or simply, R is n–stable), whenever for all s ≥ n, any unimodular sequence (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}) of elements in R is stable. It is clear, of course, that if R is n–stable, then it is m–stable for any integer m ≥ n. For more information on stable range in commutative rings, see [1] and [5]. Let J(R) denote the Jacobson radical of R. A ring R is said to be a B–ring whenever for any unimodular sequence (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}), s ≥ 2, with (a_{1}, a_{2}, ⋯, a_{s−1}) ⊄ J(R), there exists b ∈ R such that (a_{1}, a_{2}, ⋯, a_{s} + ba_{s+1}) = R. In fact, we showed in [5] that R is a B–ring if and only if for any unimodular sequence (a_{1}, a_{2}, a_{3}), a_{1}, a_{2}, a_{3} ∈ R with a_{1} ∉ J(R), there exists b ∈ R such that (a_{1}, a_{2} + ba_{3}) = R. For a detailed study on B–rings, see [4] and [5].

2. Main Results. The following lemma is a result on B–rings which can be found in [4] and a result on n–stable rings which is proved in [5]. Here for the sake of completeness, we state and give a partial proof to this lemma as follows:

\underline{Lemma 1}. Assume A ⊂ J(R) is a nonzero proper ideal of R. Then R is n–stable (respectively, a B–ring) if and only if R/A is n–stable (respectively, a B–ring).

\underline{Proof}. Here we just prove this lemma for n–stable rings. Proof for the case of B–rings which can be found in [4], is left to the reader.

\underline{Necessity Part}. Let (a_{1} + A, a_{2} + A, ⋯, a_{s} + A, a_{s+1} + A) = R/A. Hence,

1 + A = ∑_{i=1}^{s+1} a_{i}r_{i} + A

for some r_{1}, r_{2}, ⋯, r_{s}, r_{s+1} ∈ R, implies

(1 − ∑_{i=1}^{s+1} a_{i}r_{i}) ∈ A.

Thus, for some a ∈ A we get 1 ∈ (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}r_{s+1} + a). Now since R is n– stable, there exists b_{1}, b_{2}, ⋯, b_{s} ∈ R such that 1 ∈ (a_{1} + b_{1}(a_{s+1}r_{s+1} + a), ⋯, a_{s} + b_{s}(a_{s+1}r_{s+1} + a)). And now we can conclude that 1 + A ∈ (a_{1} + b_{1}r_{s+1}a_{s+1} + A, ⋯, a_{s} + b_{s}r_{s+1}a_{s+1} + A), which implies R/A is n–stable. Conversely, assume (a_{1}, a_{2}, ⋯, a_{s}, a_{s+1}) is a unimodular sequence in R. Thus, we get 1 + A ∈ (a_{1} + A, a_{2} +A, ⋯, a_{s} +A, a_{s+1} +A). Since R/A is n–stable, then 1+A ∈ (a_{1} +b_{1}a_{s+1} + A, ⋯, a_{s} +b_{s}a_{s+1} +A) for some b_{1}, b_{2}, ⋯, b_{s} ∈ R. Thus, for some a ∈ A and some X_{1}, X_{2}, ⋯, X_{s} ∈ R we have

∑_{i=1}^{s}(a_{i} + b_{s}a_{s+1})X_{i} = 1 − a

which implies (a_{1} + b_{1}a_{s+1}, ⋯, a_{s} + b_{s}a_{s+1}) = R, since 1 − a is a unit in R (recall that a ∈ A ⊂ J(R)).

\underline{Remark}. It is obvious that the necessity part of the above lemma is still true for any nonzero proper ideal A of R. For more information see [5].

\underline{Lemma 2}. For each fixed i = 1, 2 or 3,

f = (∑_{j=0}^{∞} f_{j}) ∈ T_{i}

is a unit in T_{i} if and only if f_{0} is a unit in R.

\underline{Proof}. Since for each i = 1, 2, or 3, R is a homomorphic image of T_{i} under

f = (∑_{j=0}^{∞} f_{j}) ↦ f_{0},

thus, the necessity part is clear. In the sufficient part we just give a proof for T_{3} and leave the other cases to the reader. Assume f_{0} is a unit in R and

f = ∑_{j=0}^{∞} f_{j}

is an element in T_{3}. In order to show that f is a unit in T_{3}, it is enough to find an element

g = ∑_{j=0}^{∞} g_{j}

in T_{3} such that fg = 1. By applying induction we can determine the coefficients of g as follows: f_{0}g_{0} = 1 implies g_{0} = f_{0}^{−1}, f_{0}g_{1} + f_{1}g_{0} = 0 implies g_{1} = −f_{0}^{−1}f_{1}g_{0}. Now assume we have g_{0}, g_{1}, g_{2}, ⋯, g_{k−1} and we want to determine g_{k}. From f_{0}g_{k} + f_{1}g_{k−1} +⋯+f_{k}g_{0} = 0 we have g_{k} = −f_{0}^{−1}(f_{1}g_{k−1} +f_{2}g_{k−2} +⋯+f_{k}g_{0}) and notice that here each term in parentheses is either zero or a form of degree k. Thus, g_{k} is either zero or a form of degree k and the proof by induction is complete.

We showed in [5] that the ring of formal power series R[[X_{1}, X_{2}, ⋯, X_{m}]] with a finite number of indeterminates over R is n–stable (respectively, a B–ring) if and only if R is n–stable (respectively, a B–ring). Next we generalize these results to a formal power series with any number of indeterminates.

\underline{Theorem 1}. For each fixed i = 1, 2, or 3, T_{i} is n–stable (respectively a B–ring) if and only if R is n–stable (respectively, a B–ring).

\underline{Proof}. For each i = 1, 2, or 3, let ϕ_{i} : T_{i} → R be a homomorphism of rings given by

f = (∑_{j=0}^{∞} f_{j}) ↦ f_{0}.

It is clear that any element

f = ∑_{j=0}^{∞} f_{j}

is in the kernel of ϕ_{i} (Kerϕ_{i}) if and only if f_{0} = 0. Thus, by Lemma 2 above, Kerϕ_{i} ⊂ J(T_{i}). Now by Lemma 1 above, the proof of the theorem is complete.

\underline{Remark}. By using mathematical induction and the fact that ϕ : R[[X]] → R given by f(X) ↦ f(0) is an epimorphism of rings with Ker(ϕ) ⊂ J(R[[X]]), the process of the Proof of Corollaries 2.20 and 2.22 in [5] as mentioned above, is very similar to the argument in the Proof of Theorem 1 above.

\underline{Acknowledgement}. The author wishes to thank Professor Robert Gilmer for his help on the construction of T_{3}.

\underline{References}

1. D. Estes and J. Ohm, “Stable Range in Commutative Rings,” Journal of Algebra, 7, (1967), 343–362.

2. R. Gilmer, Multiplicative Ideal Theory, Queens University, Kingston, Ontario, 1992.

3. T. W. Hungerford, Multiplicative Ideal Theory, Springer–Verlag, Inc, New York, 1974.

4. M. Moore and A. Steger, “Some Results on Completability in Commutative Rings,” Pacific Journal of Mathematics, 37 (1971), 453–460.

5. A. M. Rahimi, Some Results on Stable Range in Commutative Rings, Ph.D. Dissertation, University of Texas at Arlington, 1993.

6. O. Zariski and P. Samuel, Commutative Algebra, Vol. I, Van Nostrand Co., New York, 1958.

Amir M. Rahimi 901 Carro Dr. #4 Sacramento, CA 95825