A decomposition theorem for semiprime rings
A ring A is called an F DI-ring if there exists a decomposition of the identity of A in a sum of finite number of pairwise orthogonal primitive idempotents. We call a primitive idempotent e artinian if the ring eAe is Artinian. We prove that every semiprime F DI-ring is a direct product of a semi...
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irk-123456789-1565952019-06-19T01:28:18Z A decomposition theorem for semiprime rings Khibina, M. A ring A is called an F DI-ring if there exists a decomposition of the identity of A in a sum of finite number of pairwise orthogonal primitive idempotents. We call a primitive idempotent e artinian if the ring eAe is Artinian. We prove that every semiprime F DI-ring is a direct product of a semisimple Artinian ring and a semiprime F DI-ring whose identity decomposition doesn’t contain artinian idempotents. 2005 Article A decomposition theorem for semiprime rings / M. Khibina // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 1. — С. 62–68. — Бібліогр.: 4 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 16P40, 16G10. http://dspace.nbuv.gov.ua/handle/123456789/156595 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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A ring A is called an F DI-ring if there exists
a decomposition of the identity of A in a sum of finite number
of pairwise orthogonal primitive idempotents. We call a primitive idempotent e artinian if the ring eAe is Artinian. We prove
that every semiprime F DI-ring is a direct product of a semisimple
Artinian ring and a semiprime F DI-ring whose identity decomposition doesn’t contain artinian idempotents. |
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Article |
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Khibina, M. |
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Khibina, M. A decomposition theorem for semiprime rings Algebra and Discrete Mathematics |
| author_facet |
Khibina, M. |
| author_sort |
Khibina, M. |
| title |
A decomposition theorem for semiprime rings |
| title_short |
A decomposition theorem for semiprime rings |
| title_full |
A decomposition theorem for semiprime rings |
| title_fullStr |
A decomposition theorem for semiprime rings |
| title_full_unstemmed |
A decomposition theorem for semiprime rings |
| title_sort |
decomposition theorem for semiprime rings |
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Інститут прикладної математики і механіки НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/156595 |
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A decomposition theorem for semiprime rings / M. Khibina // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 1. — С. 62–68. — Бібліогр.: 4 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT khibinam adecompositiontheoremforsemiprimerings AT khibinam decompositiontheoremforsemiprimerings |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 1. (2005). pp. 62 – 68
c© Journal “Algebra and Discrete Mathematics”
A decomposition theorem for semiprime rings
Marina Khibina
Communicated by M. Ya. Komarnytskyj
Dedicated to Yu.A. Drozd on the occasion of his 60th birthday
Abstract. A ring A is called an FDI-ring if there exists
a decomposition of the identity of A in a sum of finite number
of pairwise orthogonal primitive idempotents. We call a primi-
tive idempotent e artinian if the ring eAe is Artinian. We prove
that every semiprime FDI-ring is a direct product of a semisimple
Artinian ring and a semiprime FDI-ring whose identity decompo-
sition doesn’t contain artinian idempotents.
1. Introduction
In this paper all rings are associative with 1 6= 0. Recall that a nonzero
idempotent e ∈ A is called local if the ring eAe is local. Obviously, every
local idempotent is primitive. The well-known Müller’s Theorem [4] gives
the following criterion for a ring A to be semiperfect:
A ring is semiperfect if and only if 1 ∈ A can be decomposed into a
sum of a finite number of pairwise orthogonal local idempotents.
For every associative ring A with 1 6= 0 we prove the theorem:
The following statements for a ring A are equivalent:
(1) the idempotent e ∈ A is local;
(2) the projective module P = eA has exactly one maximal submodule.
The following important notion used in the paper is the notion of
finitely decomposable identity ring (or for short, FDI-ring, see [2], p. 77):
2000 Mathematics Subject Classification: 16P40, 16G10.
Key words and phrases: minor of a ring, local idempotent, semiprime ring,
Peirce decomposition.
M. Khibina 63
a ring A is called an FDI-ring if there exists a decomposition of the
identity 1 ∈ A
1 = e1 + e2 + . . . + en
into a sum of finite number of pairwise orthogonal primitive idempotents
e1, . . . , en. Obviously, every semiperfect ring and every right Noetherian
ring is a FDI-ring.
We call a FDI-ring A piecewise right Artinian if all rings eiAei are
right Artinian for i = 1, . . . , n.
We prove that every semiprime FDI-ring A is a direct product of a
semisimple Artinian ring and an FDI-ring which is not piecewise right
Artinian.
The main working tool of this paper is the notion of a minor of the
ring A: Let A be a ring, P a finitely generated projective A-module which
is a direct sum of n indecomposable modules. The ring of endomorphisms
B = E(P ) of the module P is called a minor of order n of the ring A (see
[1]).
Many properties carry over from the ring to all of its minors. Follow-
ing [1] we shall say that a property Φ of a ring A is N -minoral property if
and only if all its minors whose orders are not greater than a prescribed
value N have this property Φ.
The following examples are given in [1].
Example 1.1. An Artinian ring A is semisimple if and only if for any two
indecomposable projective A-modules P1 6≃ P2, HomA(P1, P2) = 0 and
HomA(P1, P1) is a division ring. Therefore semisimplisity is a 2-minoral
property.
Example 1.2. An Artinian ring A is generalized uniserial (i.e., Artinian se-
rial) if and only if for any indecomposable projective A-modules P1, P2, P3
and for any homomorphisms ϕ1 : P1 → P3 and ϕ2 : P2 → P3, one of
the equations: ϕ1 = ϕ2x or ϕ2 = ϕ1y is solvable, where x : P1 → P2
and y : P1 → P3. Therefore, the property of being generalized uniserial
is 3-minoral.
Example 1.3. The property of being hereditary for an order Λ in a semisim-
ple k-algebra Λ̃ is 2-minoral.
On other hand, an analogous notion is defined in the paper [3]:
Let C be a class of rings, and P a property that rings in C may or may
not have. We say that P is k-determined in C if a ring Λ in C has P if
and only if all eΛe have P, for e a sum of at most k pairwise orthogonal
primitive idempotents of Λ.
The following two properties are proved in [3].
64 A decomposition theorem for semiprime rings
Proposition 1.4. The property of being left serial is three-determined in
the class of Artinian rings.
Proposition 1.5. The property of being hereditary is two-determined in
the class C of orders over complete discrete valuation rings.
2. Projective modules
Let M be an A-module. We set rad M = M , M has no maximal
submodules, and otherwise, rad M denotes the intersection of all maximal
submodules of M . We write R = R(A) = rad AA, s the Jacobson radical
of A.
The following proposition is well-known (see, for example, [2], Propo-
sition 4.2.10, p. 115).
Proposition 2.1. If P is a nonzero projective A-module, then rad P =
P · rad A 6= P .
Theorem 2.2. Suppose that P = eA (e2 = e 6= 0) has exactly one
maximal submodule. Then the idempotent e is local. Conversely, if e is a
local idempotent and P = eA, then PR is the unique maximal submodule
of P .
Proof. Suppose that P = eA has exactly one maximal submodule M .
Then by Proposition 2.1 M = PR. For any ϕ : P → P either Imϕ = P
or Imϕ ⊆ PR.
In the first case, since P is projective, we have P ≃ Imϕ ⊕ Ker ϕ
which implies Ker ϕ = 0. So, ϕ is an automorphism.
In the second case ϕ is non-invertible. Obviously, all non-invertible
elements of HomA (P, P ) ≃ eAe form an ideal and therefore the ring eAe
is local.
Conversely, let e be a local idempotent of the ring A and π : A → Ā
be the natural epimorphism of A into Ā = A/R (R is the Jacobson
radical of A). We denote π(a) = ā. Suppose 1 6= e. We have 1 = e + f
and ef = fe = 0. Obviously, f̄ Ā is a proper right ideal in Ā. So, it is
contained in a maximal right ideal Ĩ if Ā. We will show that ēĀ∩ Ĩ = 0,
otherwise (ēĀ ∩ Ĩ)2 6= 0.
Since Ā is a semiprimitive ring then (ēĀ ∩ Ĩ)2 = 0. There exists
ēā ∈ Ĩ and ēāēā 6= 0. So, ēāē 6= 0. Since eAe is a local ring and
rad (eAe) = eRe, then ēĀē is a division ring. Therefore, there is an
element ēx̄ē ∈ ēAē such that ēāēx̄ē = ē and ē ∈ Ĩ. Thus 1̄ ∈ Ĩ. We
get a contradiction. Therefore ēĀ ∩ Ĩ = 0 and Ā = ēĀ ⊕ Ĩ. Since Ĩ
is maximal ideal in Ā then ēĀ is simple and PR is the unique maximal
submodule in P = eA.
M. Khibina 65
Let A be an FDI-ring with the following decomposition of identity
1 ∈ A:
1 = e1 + . . . + en.
We may assume that all rings eiAei are local for i = 1, . . . , k and the
rings ejAej are non-local for j = k + 1, . . . , n. Put e = e1 + . . . + ek and
f = 1 − e. Let eAf = X, fAe = Y and
A =
(
eAe X
Y fAf
)
(∗)
be the corresponding two-sided Peirce decomposition of A. By Müller’s
Theorem the ring eAe is semiperfect.
We shall call the decomposition (∗) standard two-sided Peirce decom-
position of a FDI-ring A.
3. Piecewise right Artinian semiprime rings are semisim-
ple Artinian
Recall that a ring A is called semiprime if A does not contain nonzero
nilpotent ideals. We shall need the following lemma.
Lemma 3.1. Let e be a nonzero idempotent of a ring A. For any nilpo-
tent ideal I of the ring eAe there exists a nilpotent ideal I of A such that
eĨe = I.
Proof. Let f = 1 − e and Ĩ = I + IeAf + fAeI + fAeIeAf . It is clear
that Ĩ is the nilpotent ideal.
Corollary 3.2. Let e be a nonzero idempotent of a semiprime ring A.
Then the ring eAe is semiprime.
Definition 3.3. A ring A with the Jacobson radical R is called semipri-
mary if A/R is semisimple Artinian and R is nilpotent.
Theorem 3.4. A piecewise right Artinian ring A is semiprimary.
Proof. Obviously, A is semiperfect. Let 1 = e1 + . . . + en be the decom-
position of 1 ∈ A into the sum of a finite number of pairwise orthogonal
local idempotents. Let R = rad AA be the Jacobson radical of A. Then
eiRei = rad (eiAei) is either zero or nilpotent. By induction on n it is
easy to see, that R is a nilpotent ideal. So, A/R is semisimple Artinian
and A is semiprimary.
66 A decomposition theorem for semiprime rings
Example 3.5. Let
A =
{(
α β
0 α
)
|α ∈ Q, β ∈ R
}
.
Obviously, A is a local semiprimary ring which is not right or left Artinian.
This example shows that the converse of Theorem 3.4 is not true.
Proposition 3.6. The property of being semiprimary is 1-minoral in the
class of FDI-rings.
Proof is analogous to the proof of Theorem 3.4.
Theorem 3.7. A semiprimary semiprime ring A is semisimple Artinian.
Proof. By definition of a semiprime ring we have that R = 0 and A is
semisimple Artinian.
Corollary 3.8. Piecewise right Artinian semiprime ring is semisimple
Artinian.
4. A decomposition theorem for semiprime rings
Recall that a ring A is said decomposable if A is a direct product of two
rings. Otherwise a ring A is called indecomposable.
Definition 4.1 ([2], p.74). A ring A is called finitely decomposable (or,
for short, FD-ring) if it decomposes into a direct product of a finite num-
ber of indecomposable rings.
Proposition 4.2 ([2], Corollary 2.5.15, p.77). Any FDI-ring is an FD-
ring.
Obviously, we have the following Proposition.
Proposition 4.3. Let A be a semiprime FDI-ring. Then A is a finite
direct product of semiprime indecomposable FDI-rings.
We fix the decomposition of the identity 1 ∈ A (where A is an inde-
composable semiprime FDI-ring) in a sum
1 = e1 + . . . + en
of a finite number of pairwise orthogonal primitive idempotents e1, . . . , en.
Definition 4.4. A primitive idempotent e shall be called artinian if the
ring eAe is Artinian.
M. Khibina 67
Theorem 4.5. Let A be an indecomposable semiprime FDI-ring. The
ring A is isomorphic to the ring Mn(D) if and only if ei ∈ A is artinian
for some i..
Proof. Suppose that ek is artinian and ej is not artinian for j > k.
Consider the following minor of the second order
Bk,j =
(
ekAek ekAej
ejAek ekAek
)
for k > j. Obviously, ekAek is a division ring. Denote by Rk,j the
Jacobson radical of Bk,j . Let P
(k,j)
1 = ekBk,j and P
(k,j)
2 = ejBk,j . By
Theorem 2.2 P
(k,j)
1 Rk,j is the unique maximal submodule of P
(k,j)
1 . So,
we have:
P
(k,j)
1 Rk,j ⊂ (0, ekAej) ⊂ P
(k,j)
1 .
Then each element ekaej ∈ ekAej defines a homomorphism ϕk : P
(k,j)
2 →
P
(k,j)
1 such that Imϕk,j ⊆ P
(k,j)
1 Rk,j , i.e., ekaejejha1ek = 0 for any
a, a1 ∈ A. Therefore,
J =
(
0 ekAej
ejAek ejAek
)
is a nilpotent ideal in Bk,j . By Lemma 3.1 ekAej = 0 and ejAek = 0.
Let h1 = e1 + . . . + ek and h2 = ek+1 + . . . + en, X = hAh2 and
Y = h2Ah1. Let
A =
(
h1Ah1 X
Y h2Ah2
)
be the corresponding two-sided Peirce decomposition. As above we have
X = 0 and Y = 0. It follows from indecomposability of A that A is the
piecewise Artinian ring and by Theorem 3.7 A ≃ Mn(D), where Mn(D)
is a ring of all n × n-matrices with elements in a division ring A. The
converse assertion is obvious.
Corollary 4.6 (A decomposition theorem for semiprime rings). Every
semiprime FDI-ring is a direct product of a semisimple Artinian ring
and a semiprime FDI-ring whose identity decomposition doesn’t contain
artinian idempotents.
References
[1] Drozd, Yu.A., Minors and reduction theorems, Coll. Math. Soc. J.Bolyai, v.6,
(1971), pp. 173-176.
68 A decomposition theorem for semiprime rings
[2] Gubareni, N.M. and Kirichenko, V.V., Rings and Modules. - Czestochowa, 2001.
[3] Gustafson, W.H., On hereditary orders, Comm. in Algebra, 15(1&2) (1987), pp.
219-226.
[4] Müller, B., On semi-perfect rings, Ill. J.Math., v.14, N3 (1970), pp. 464-467.
Contact information
M. Khibina In-t of Engineering Thermophysics, NAS,
Ukraine
E-Mail: marina_khibina@yahoo.com
Received by the editors: 27.09.2004
and in final form 21.03.2005.
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