Some Properties of Noncommutative rings of Hurwitz Series
Susan F. El-Deken
Department of Mathematics, Faculty of Science, Helwan University, Ain Helwan, 11790, Cairo, Egypt.
E-mail: eldekens@yahoo.com
Abstract.
This paper investigates the structure of the class of Hurwitz series over
noncommutative rings. Generalizing theorems of commutative rings of Hurwitz series,
we prove the noncommutative situations of these results. We focus on special class,
namely Homogeneous Semilocal rings of Hurwitz series and show that many of the
properties of local rings of Hurwitz series can be hold for homogeneous semilocal rings.
Homogeneous semilocal rings that is, the ring R with R J (R) simple artinian.
Introduction
In an earlier paper [6] Keigher studied the properties of formal power series ring
and its categorical properties. In the sequel paper [5] Keigher introduced an extension of
formal power series ring which called the ring of Hurwitz series, and examines its
structure and applications, especially to the study of differential algebra. The Hurwitz
series rings are similar to formal power series rings except that binomial coefficients are
introduced at each term in the product. While there are many studies of these rings over a
commutative ring, very little is known about them over a noncommutative ring. In the
present paper we study Hurwitz series over a noncommutative ring with identity,
examine its structure and properties. We focus on special class, namely, Homogeneous
Semilocal rings, that is, the ring R with R J (R) simple artinian [3].
Throughout this paper, all rings are associative with identity. A domain is a
(commutative or noncommutative) ring without zero divisors. We use J R  and U R to
denote the Jacobson radical and group of units of R , respectively. A ring R is called
local ring (homogeneous semilocal ring, semilocal ring) if R J (R) is a division ring
( R J (R) simple artinian, R J (R) semisimple ring). M n (R) denotes the ring of n n
matrices over R .
1
1. Overview of basic definitions
The definition of Hurwitz series originally allowed the ring to be noncommutative, but most authors restrict them to be commutative, therefore all of the basic
definitions are still true under the restriction that the ring is noncommutative. The
Hurwitz series over any ring R (not necessary commutative) is defined as a function
from N  R and denoted by HR . The elements of the ring HR of Hurwitz series
over R are aHR   an X n for all n  N = {0, 1, 2, 3,…}. The formula for addition in HR
nN
is defined component-wise, so that,
aHR  bHR   an X n   bn X n   (an  bn ) X n , for all n  N .
nN
nN
nN
The multiplication is defined as the Cauchy product given by
aHRbHR  (  an X n )(  bn X n )   (  Cin aib j ) X n , for all n  N ,
n N
n N
n N i  j  n
the sum of all aib j where i, j range over all pairs of elements of N which sum to n .
Cin  n!/(i!(n  i )!) denote the binomial coefficient defined for all i, n  N with n  i .
With these two operations the Hurwitz series ring HR is a noncommutative ring with
identity containing R .The zero element in HR is 0HR  (0,0,0,...) , the sequence with all
terms 0, and the identity is 1HR  (1,0,0,...) , the sequence with 0th term 1 and nth term 0
for all n  1 .
In the following section we investigate the structure of Homogeneous semilocal
ring.
2. Structure of Homogeneous semilocal ring
In noncommutative ring, local ring have a unique maximal left ideal (unique
maximal right ideal) equivalent to have a unique maximal two-sided ideal but the
converse is not true, i.e. the ring having a unique maximal two sided ideal is not
necessary local. One of the extension classes of local ring is homogeneous semilocal ring
which has a unique maximal two-sided ideal. The Jacobson radical J (R) of a
homogeneous semilocal ring R coincides with the unique maximal two sided ideal. A
ring R is said to be a homogeneous semilocal ring if R / J ( R ) is simple Artinian. In [3]
Facchini was shown that these rings share a number of properties with local rings. For
instance, the Jacobson radical J (R) of a semilocal homogeneous ring R is its unique
maximal two-sided ideal, that is, every proper two-sided ideal of R is contained in J (R) .
A ring R is said to be right primitive if R has a faithful simple right module. For a
ring R , an ideal P of R is called a right primitive ideal if the factor ring R / P is a right
primitive ring. The intersection of all the right primitive ideals of R coincides with the
intersection of all the left primitive ideals of R and is equal to the Jacobson radical J (R) .
We now prove some properties of homogeneous semilocal rings.
2
Proposition 2.1. If R is a homogeneous semilocal ring, then the Jacobson radical J (R) is
the only primitive ideal.
Proof. Let P be a right primitive ideal of R . Then there exists some maximal right ideal
of R contains P as a largest two sided ideal. But R has a unique maximal two sided
J ( R )  P is the only primitive ideal in a homogeneous semilocal ring R .
ideal J (R)
Proposition 2.2. If R is a homogeneous semilocal ring, then the primitive factor ring
R / J ( R ) is artinian and also every prime factor ring of R is left artinian.
Proof. If R is a homogeneous semilocal ring, then the factor ring R / J ( R ) is simple
artinian.
Proposition 2.3. If R is a homogenous semilocal ring, then every maximal one-sided
ideal contains a maximal two-sided ideal.
Proof. Let M R be a maximal right ideal of R . Then there exists a right primitive ideal
of R contained in M R ; hence J (R) is the only primitive ideal in R . Then every maximal
one-sided ideal contains a maximal two-sided ideal.
Proposition 2.4. If R is a homogenous semilocal ring, then R is a Dedekind-finite.
Proof. It is evident from ([8] prop. 20.8.).
3. Hurwitz series over homogeneous semilocal ring
The aim of this section is studying the ring of Hurwitz series over
noncommutative rings, especially, homogeneous semilocal ring. We further generalizing
theorems of commutative rings of Hurwitz series to noncommutative situations of these
results.
For any ring R , the only units in the polynomial ring R[x] are the units of R , while
the formal power series R[[x]] has many units. A necessary and sufficient condition for an
element in the ring of power series to be a unit is that its constant term is invertible in R .
In 1975 Keigher proved the similar result holds in Hurwitz series ring. Keigher's works
has been focused on commutative case; in the present paper we extend his results in
noncommutative case. The following proposition determines the invertible elements in
Hurwitz series ring over any ring.
Proposition 3.1. [6] For any ring R the Hurwitz series (an )  (a0 , a1, a2 ,...) is a unit in
HR if and only if a0 is a unit in R .
The following example illustrates unusual behavior in rings of Hurwitz series
3
Example 3.2. [5]. Let R  Z 2 denote the field. Let aHR  (0, 1R ,0,0,0,. . .) is in HR be the
Hurwitz series over R with first term 1 (recall that terms are indexed starting with 0) and
all other terms 0. Then x 2  0 in HR , so HR may fail to be a domain, even if R is a field.
The following result generalize Keigher’s result [5, Corollary 2.8], and also Ali
Benhissi result [1, Proposition1.1 ].
Corollary 3.3. The Hurwitz series HR is a domain if and only if R is a domain and
char ( R)  0.
Proof.
 Since R  HR , then R is a domain. Suppose that mR  0 . Then
X  X m 1  C1m X m  m1R X m  0.
 Suppose R is a domain. If aHR  (an )  (a0 , a1 , a2 ,...) be in HR , with aHR  0 .
Then ai  0 for some i  0 ; Now suppose bHR   bn X n is in HR and that aHRbHR  0 .
n0
Recall that aHRbHR   (  C a b j ) X , so that for all n we have
nN i  j  n
n
i i
n

i  j n
Cin ai b j  0 .
We will now show by induction that b j for all j .
First, let n  k . Since

i  j n
Cin ai b j  0 and ai  0 for all
, we have ak b0  0 . Since R
is a domain, b0  0 .
Suppose
Since

that
for
some
s  0, b j  0 when
C a b j  0 , and ai  0 when 0 
i  j  k  s 1
n
i i
0 j  s.
Let n  k  s  1.
and b j  0 when 0  j  s , we have
ak bs 1  0 . Since R has no zero divisors and ak  0 , bs 1  0 . Let k be minimal in this set.
Thus bn  0 for all n , and we have bHR  0 . Thus aHR has no right zero divisors.
Similarly, it has no left zero divisors, and thus HR is a domain.
The locality has been studied on Hurwitz series rings over commutative local ring
by Keigher in [6]. In [1], Benhissi gave a more general result. The connection was made
precise in the following results.
Proposition 3.4. [6] If R is a commutative local ring with maximal ideal M R , then the
Hurwitz series ring is also a local ring, with maximal ideal M HR .
Corollary 3.5. [1] Let R be a commutative ring. The Hurwitz series ring HR is local
(resp. quasi local) if and only if R is local (resp. quasi local).
4
The above results can be generalized in the noncommutative case. In this paper
we show that the homogeneous semilocality can be transferred directly from R to HR .
{Local rings}  {Homogeneous semilocal rings}
Proposition 3.6. Let R be a homogeneous semilocal ring with maximal ideal M R . Let
 R : HR  R be a natural ring homomorphism defined for any (an )  HR ,  R (an )  a0 .
Then HR is a homogeneous semilocal ring with maximal ideal
M HR   n1 ( M R )  {( a0 , a1 , a2 ,...) | a0  M R } .
Proof.
Since R is a homogeneous semilocal ring with maximal ideal J ( R)  M R , then
HR / J ( HR )  R / J ( R) , which is simple artinian ring. Therefore, R is also a
homogeneous semilocal ring and the Jacobson radical is a maximal ideal of Hurwitz
series ring HR which is M HR   n1 ( M R )  {( a0 , a1 , a2 ,...) | a0  M R } .
Reference
[1] A. Benhissi, Ideal structure of Hurwitz series rings. Contrib. Algebra Geom. 48(1),
251–256 (2007)
[2] S. Chen, J. Q. Li, On rings whose primitive factor rings are left artinian, Southeast
Asian Bull. Math. 25 (2001), no. 1, 23–30.
[3] R. Corisello and A. Facchini, Homogeneous semilocal rings, Comm. Algebra 29
(2001), no. 4, 1807–1819.
[4] A. Facchini, Module Theory: Endomorphism Rings and Direct Sum Decompositions
in Some Classes of Modules, Progress in Math., Vol. 167, Birkhauser Boston, 1998.
[5] W. F. Keigher, On the ring of Hurwitz series, Comm. Algebra 25 (1997), 1845-1859.
[6] W.F. Keigher, Adjunctions and comonads in differential algebra, Pacific J. Math.
59(1975), 99-112.
[7] T.Y.Lam, A First Course in Non-Commutative Rings, GTM 131, Springer, Berlin,
1991.
[8] T. Y. Lam, Exercises in Classical Ring Theory, Springer, Berlin, 1995
[9] J. Lambek, Lectures on rings and modules, Blaisdell, London, 1966.
[10] Y. Lee and C. Huh, On rings in which every maximal one-sided ideal contains a
maximal ideal, Comm. Algebra 27 (1999), no. 8, 3969–3978.
5