J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
170Ae174.
ISOMORPHISM BETWEEN ENDOMORPHISM RINGS
OF MODULES OVER A SEMI SIMPLE RING
Hery Susanto1 .
Santi Irawati1 .
Indriati Nurul Hidayah1 , and Irawati2 Mathematics Department.
Universitas Negeri Malang.
Jalan Semarang 5.
Malang 65145.
Indonesia fmipa@um.
id, santi.
fmipa@um.
fmipa@um.
Mathematics Department.
Institut Teknologi Bandung.
Jalan Ganesha 10.
Bandung 40132.
Indonesia irawati@math.
Abstract.
Our question is what ring R which all modules over R are determined, up to isomorphism, by their endomorphism rings? Examples of this ring are division ring and simple Artinian ring.
Any semi simple ring does not satisfy this property.
We construct a semi simple ring R but R is not a simple Artinian ring which all modules over R are determined, up to isomorphism, by their endomorphism rings.
Key words and Phrases: Modules, division ring, simple Artinian ring, semi simple Introduction Let M.
N be modules over a ring R and set of all R-homomorphisms from M to N is written HomR (M.
N ).
Then HomR (M.
N ) is Abelian group over addition of mapping.
Moreover.
EndR (M ) = HomR (M.
M ) is a ring over addition and composition of mapping, and called endomorphism ring of M .
In general if two modules are isomorphic then their endomorphism rings are isomorphic.
The converse is not true.
The Baer-Kaplansky theorem states that two Abelian torsion groups are isomorphic if and only if their endomorphism rings are isomorphic .
Ivanov got same result with the Baer-Kaplansky theorem.
Let R be the upper triangular ring over a division ring and F be the category of R-modules which have a summand isomorphic to R.
Then modules in F are determined, up isomorphism, by their endomorphism rings .
Our question is what ring 2020 Mathematics Subject Classification: 16D10 Received: 01-11-2018, accepted: 01-04-2019.
Isomorphism between endomorphism rings of modules over a semi simple ring which modules in R-MOD are determined, up to isomorphism, by their endomorphism rings? Examples of this ring are division ring and simple Artinian ring.
Any semi simple ring does not satisfy this property.
We construct a semi simple ring R but R is not a simple Artinian ring which modules in R-MOD are determined, up to isomorphism, by their endomorphism rings.
In this paper a ring will be a ring with unity and all modules will be nonzero unital right module except if in special case.
Definition of ring, module, and others which are used in this paper refer to .
IP-isomorphism In Baer-Kaplansky theorem and Ivanov result, isomorphism between endomorphism rings of two modules always Aypreserve indecomposable direct summandAy, which called an IP-isomorphism.
For a class of modules which have a decomposition M = OiiOOI Mi with property that every indecomposable direct summand of M is contained in the sum of finite number of the Mi .
e say that M has the finite embedding propert.
, an IP-isomorphism between endomorphism rings of modules give an isomorphism between the modules.
An element e in a ring R is called idempotent if e2 = e.
Two idempotents e1 and e2 in R are called orthogonal if e1 e2 = 0 = e2 e1 .
An idempotent e 6= 0 in R is called primitive idempotent if e can not be a sum of two nonzero orthogonal Let M be a module over a ring R and e an idempotent in EndR (M ).
Then 1 Oe e is also idempotent in EndR (M ).
Moreover, e and 1 Oe e orthogonal and M has decomposition M = eM Oi .
Oe .
M .
Then the direct summand eM is indecomposable if and only if e is a primitive idempotent in EndR (M ) (.
Lemma Let given M and N be modules over a ring R, and i : M OeIe N any Risomorphism.
The mapping which defined : EndR (M ) OeIe EndR (N ), (O) = iOiOe1 is a ring isomorphism.
Furthermore, for any idempotent e in EndR (M ) satisfies .
N = ieiOe1 N = .
Oe1 N ) = .
M = i.
M ).
Therefore .
eM : eM OeIe .
N is a R-isomorphism.
So eM O = .
N , for all e idempotent in EndR (M ).
In general, if : EndR (M ) OeIe EndR (N ) is any ring isomorphism and e is any primitive idempotent in EndR (M ) then .
is also a primitive idempotent in EndR (N ) but it must not eM O = .
N .
Let M and N be modules over a ring R.
A ring isomorphism I : EndR (M ) OeIe EndR (N ) SUSANTO.
IRAWATI.
HIDAYAH.
IRAWATI
is called IP-isomorphism if I.
N O = eM , for all primitive idempotents e in EndR (M ).
The following proposition gives property that an IP-isomorphism between endomorphism rings will give module isomorphism.
Proposition 2.
Let M and N be modules over a ring R where M has the finite imbedding property with respect to a decomposition into indecomposable direct summands and N be generated by indecomposable direct summands.
Then M and N are isomorphic if and only if there is an IP-isomorphism between EndR (M ) and EndR (N ).
Proof.
See .
Proposition 1.
Main Results Let Zp and Zq be fields of integer number modulo p and q, respectively, where p and q are different prime numbers.
Let Zp 0 a 0 0 R = 0 Zq Zq = 0 b c a OO Zp , b, c, d, e OO Zq 0 Zq Zq 0 d e Then R is a semi simple ring R but R is not a simple Artinian ring.
As a left modul over itself.
R has decomposition over simples modules as follows.
= 0 Oi Zq Oi Zq = 0 Oi Zq .
Proposition 3.
Let M and N be modules over the ring R in .
Then every ring isomorphism between EndR (M ) and EndR (N ) is an IP-isomorphism.
Proof.
Let I : EndR (M ) Ie EndR (N ) be any ring isomorphism and e be any primitive idempotent in EndR (M ).
We will be shown that eM O = I.
N .
Because e primitive idempotent in EndR (M ) then eM O = 0 or eM O = Zq and I.
is idempotent primitive in EndR (N ).
Suppose that eM O = 0 then Zp O = EndR .
M ) O = eEndR (M )e O = I.
EndR (N )I.
O = EndR (I.
N ).
Isomorphism between endomorphism rings of modules over a semi simple ring Suppose that I.
N O 6 0 then I.
N O = Zq .
So that EndR (I.
N ) O
Zq O 6 Zp .
So I.
N O
= 0 O
= eM .
In the case of eM O = Zq , with proof as before, we have the same results.
So I is IP-isomorphism.
Corollary 3.
Let M and N be modules over the ring R in .
Then M O = N if and only if EndR (M ) O = EndR (N ).
Proof.
Because ring isomorphism between EndR (M ) and EndR (N ) always in the form of IP-isomorphism and decomposition of semi simple modules fulfills the insertion properties so that according to the Proposition 2.
1 obtained M O = N.
As a generalization of the results above, the following propositions are obtained.
Proposition 3.
Let R be a semi simple ring and has decomposition RO = I1 1 Oi A A A Oi Is.
s ) O Il if and only if k = l, and where Ik is a simple ideal.
Ik = RO = Mn (D1 ) Oi A A A Oi Mn (Ds ) .
) .
where Dk is a division ring.
Then the following properties are equivalent.
Every ring isomorphism I : EndR (M ) OeIe EndR (N ), where M and N are modules over R, is a IP-isomorphism.
EndR (Ik ) EndR (Il ), k 6= l.
Dk Dl , k 6= l.
Corollary 3.
Let R be a semi simple ring which has decomposition .
and EndR (Ik ) EndR (Il ), k 6= l.
Then for M and N modules over R, we have M O if and only if EndR (M ) O = EndR (N ).
Acknowledgement.
This research is supported by Dana PNBP Universitas Negeri Malang 2018 organized by LP2M Universitas Negeri Malang.
The authors thank the anonymous referees for their valuable suggestions which let to the improvement of the manuscript.
REFERENCES