Abstract In this paper I have tried to find some of the results on endomorphism rings of small pseudo projective modules. Mathematics Subject Classification: 16D4

P C Bharadwaj

Ritu Jaiswal

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Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 13, 603 - 608
Endomorphism Rings of Small
Pseudo Projective Modules
Ritu Jaiswal
Department Of Mathematics
Banaras Hindu University
Varanasi-221005, India
ritu11 bhu@yahoo.com
P. C. Bharadwaj
Department Of Mathematics
Banaras Hindu University
Varanasi-221005, India
drpcbharadwaj@yahoo.co.in
Abstract
In this paper I have tried to find some of the results on endomorphism
rings of small pseudo projective modules.
Mathematics Subject Classification: 16D40
Keywords: Small pseudo projective modules, small pseudo stable sub-
module
1 Introduction
Throughout this paper the basic ring R is supposed to be ring with unity and
all modules are unitary left R-modules.
Let M be an R-module, a submodule K of M is said to be small in M if
K +L = M ⇒ L = M for any submodule L ⊆M . An R-module M is said to
be hollow if all proper submodules of M are small in M . An R-module M is
said to be small quasi projective if for any module A, with small epimorphism
g : M → A and homomorphism f : M → A there exists an h ∈ End(M) such
that f = goh. An R-module M is said to be small pseudo projective if for any
module A, with small epimorphism g : M → A and epimorphism f : M → A
there exists an h ∈ End(M) such that f = goh. A ring R is called regular
604 R. Jaiswal and P. C. Bharadwaj
(in the sense of Von- Neumann) if for each r ∈ R there exists x ∈ R such
that r = rxr. The jacobson radical J(M), of a module M , is the intersection
of all maximal submodules of M . An R-module M is called local if it has
a unique maximal submodule which contains every proper submodules of M.
The socle of an R module M denoted by Soc(M) is defined as intersection
of essential submodules of M . Two module epimorphisms f, g : P → M are
right equivalent if f = goh for some automorphism h of P . An R module M
is called π-projective if for all submodules U and V of M with U + V = M ,
there exists f ∈ S with Imf ⊆ U and Im(1 − f) ⊆ V . A submodule N
of an R-module M is said to be small pseudo stable if for any epimorphism
f : M → A and any small epimorphism g : M → A with N ⊆ Kerg ∩Kerf ,
there exists h ∈ End(M) such that f = goh then, h(N) ⊆ N . A module M is
called a duo module if every submodule of M is fully invariant.
2 Main Results
Proposition 1. Let M be any small pseudo projective hollow module. Then
every epimorphism in End(M) is an automorphism.
Proof: Let g : M → M be any epimorphism then we have Kerg = M . So,
Kerg is a proper submodule of M . As M is hollow, g is a small epimorphism,
by small pseudo projectivity of M , IM can be lifted to a homomorphism h :
M →M such that goh = IM .
⇒ h is one-one.
Let m ∈ M then as g is onto there exists an element n ∈ M such that
m = g(n) ⇒ g(n − h(m) = 0 ⇒ n − h(m) ∈ Kerg ⇒ n ∈ Kerg + h(m)
⇒ M ⊆ Kerg + Imh. So, we have M = Kerg + Imh ⇒ M = Imh, since
M is hollow. Thus h is onto and so h is an automorphism ⇒ h−1 = g is an
automorphism.
Proposition 2. (a)If S is the endomorphism ring of a small quasi projective
hollow module M then S is local.
(b)If S is the endomorphism ring of a small pseudo projective hollow module
M then S is local.
Proof: Follows from [1, Theorem 1.14]
Proposition 3. Let M be any pseudo projective module and End(M) denotes
the endomorphism ring of M . Then if α(M) ⊆⊕ M for every α ∈ End(M)
then kerα ⊆⊕ M .
Proof: Follows from [7, Proposition 8].
Endomorphism rings of small pseudo projective modules 605
Proposition 4. Let M be any pseudo projective module and End(M) denotes
the endomorphism ring of M . Then if α(M) ⊆⊕ M for every α ∈ End(M)
then End(M) is regular.
Proof: Follows from [7, Proposition 10].
Corollary 4.1: Endomorphism ring of a completely reducible pseudo projec-
tive module is regular.
Proposition 5. Let M be a small pseudo projective hollow module S denotes
the endomorphism ring of M , J(S) denotes the jacobson radical of S then
(a) J(S) = {α ∈ S|Imα is small in M}
(b)J(S) ⊆ Hom(M,J(M))
(C)S/J(S) is Von-neumann regular ring.
Proof:Follows from [1, Theorem 1.15]
Proposition 6. Let M be a small pseudo projective hollow module and K be
any small submodule of M then for any automorphism g ∈ Aut(M/K) there
exists an automorphism h ∈ Aut(M) such that g(m+K) = h(m)+K ∀m ∈M .
Proof: Let K be any small submodule of M and ν : M → M/K be any
natural map, and g : M/K → M/K be any automorphism in Aut(M/K).
Then by small pseudo projectivity of M ∃ h ∈ End(M) such that goν = νoh
i.e. goν(m) = νoh(m)∀m ∈ M ⇒ g(m +K) = h(m) +K ∀m ∈ M . Then by
[5, Proposition 4] h is an epi-endomorphism. By Proposition 1 we get h is an
automorphism.
Proposition 7. Let M be a small pseudo projective hollow module then for
any α ∈ End(M) and any small submodule K of M with α(M) + K = M
and α−1(K) = K there exists β ∈ End(M) such that β(M) ⊆ K and α+ β ∈
Aut(M).
Proof: Suppose α ∈ End(M) and K is any small submodule of M satisfying
α(M) + K = M and α−1(K) = K. Let f : M → M/K be the natural
map. Now we have Ker(foα) = α−1(Kerf) = α−1(K) = K = Kerf . Thus,
Ker(foα) = Kerf . Now, α(M) + K = M ⇒ α(M) = M ⇒ α is onto
and therefore foα is onto. So by [2, Theorem 3.6] ∃ an automorphism g ∈
End(M/K) 
 gof = foα. So by assumption there exists h ∈ Aut(M) such
that g(m + K) = h(m) + K ⇒ g(M/K) = foh(M) ⇒ gof(M) = foh(M)
⇒ gof = foh ⇒ foα = foh ⇒ f(h − α) = 0. Let β = h − α. We have
β(M) ⊆ K. Also α + β = h is an automorphism in Aut(M).
606 R. Jaiswal and P. C. Bharadwaj
Proposition 8. Let M be a small pseudo projective hollow module then any
pair of small epimorphisms from M to any module N are right equivalent if for
given any α ∈ End(M) and any small submodule K of M with α(M)+K = M
there exists β ∈ End(M) such that β(M) ⊆ K and α + β ∈ Aut(M).
Proof: Suppose f, g : M → N are small epimorphism. By small pseudo
projective of M there exists α ∈ End(M) such that f = goα. Since f is
epimorphism we have α(M) + Ker(g) = M then by assumption there exists
β ∈ End(M) such that α + β ∈ Aut(M) and β(M) ⊆ K. So g(α + β) =
goα+ goβ = goα = f . So, f and g are right equivalent.
Proposition 9. Let M be a duo and small pseudo projective module. Let S
denotes the endomorphism ring of M and T = {α ∈ S|Imα is small in M}.
Then for every f ∈ T , Imf is a small pseudo stable submodule of M .
Proof: Let f ∈ T then Imf is a small submodule of M . Let g : M/Imf →
A be a small epimorphism, ψ : M/Imf → A be an epimorphism and ν :
M → M/Imf be the natural map. Then Kerν = Imf is a small submodule
of M ⇒ ν is a small epimorphism. Now, Imf ⊆ Ker(goν) ∩ Ker(ψoν),
since Kerν = Imf ⇒ ν(Imf) = 0 ⇒ g(ν(Imf)) = 0 ⇒ Imf ⊆ Ker(goν).
Similarly Imf ⊆ Ker(ψoν)). By small pseudo projectivity of M there exists
h ∈ End(M) such that ψoν = goνoh. We have, h(Imf) ⊆ Imf , since M is
duo and Imf ⊆M . So, Imf is a small pseudo stable submodule of M .
Proposition 10. Let M be a duo and small pseudo projective hollow module.
Let S denotes the endomorphism ring of M and J(S) denotes the jacobson rad-
ical of M . Then for every f ∈ J(S), Imf is a small pseudo stable submodule
of M .
Proof: By Proposition 5(a), we have T = J(S). Rest of the proof follows
from Proposition 9.
Proposition 11. Let M be a small pseudo projective module if S is local and
M is π- projective then M is hollow.
Proof: Let U and V be submodules of M such that U + V = M . As M is π-
projective there exists f ∈ S such that Imf ⊆ U and Im(1 − f) ⊆ V . Now
S is local so, f ∈ S ⇒ either f or (1 − f) is invertible. Now f is invertible
⇒ ∃ g ∈ S 
 fog = IM ⇒ f is onto and so Imf = M ⇒ U = M . Thus V
is small. Similarly we can show that when (1 − f) is invertible then V = M
⇒ U is small, and therefore M is hollow.
Proposition 12. Let M be a small pseudo projective D2 module. Then M is
S.F.
Proof: Follows from [5, Proposition 3]
Endomorphism rings of small pseudo projective modules 607
Proposition 13. Let M be a small quasi projective duo module. If S =
End(M), is local, then M is not supplemented.
Proof: Suppose that M is supplemented and A is any submodule of M . Let
B be supplement of A in M then we have M = A + B and A ∩ B is small
in M . Let 0 = s(M) = A and 0 = t(M) = B, s, t ∈ S. Define the map
f : M = (s + t)(M) → M/(A ∩ B) such that f(s + t)(m) = s(m) + (A ∩ B).
For any m,m
′ ∈ M , (s + t)(m) = (s + t)(m′) implies that s(m − m′) =
t(m
′ −m) ∈ A ∩ B. So s(m) + (A ∩ B) = s(m′) + (A ∩ B). Thus f is well
defind and f is also an R-homomorphism. Let ν : M → M/(A ∩ B) is the
natural map. By small quasi projectivity of M , there exist g ∈ S such that
νog = f . We have νog(s + t))(m) = f(s + t)(m) = s(m) + (A ∩ B). Then,
g(s+ t))(m) + (A∩B) = s(m) + (A∩B) ⇒ ((1− g)os− got)(M) ⊆ (A∩B).
Since S is local, g or (1 − g) is invertible. If (1 − g) is invertible we have,
(s − (1 − g)−1ogot)(M) ⊆ (1 − g)−1(A ∩ B) ⊆ (A ∩ B). Now A ⊆ (s − (1 −
g)−1ogot)(M) ⊆ (A∩B). ThenA ⊆ (A∩B), which is a contradiction. Similarly
if g is invertible we have B ⊆ (g−1o(1 − g)os − t) ⊆ g−1(A ∩ B) ⊆ (A ∩ B).
Then B ⊆ (A∩B), that is also a contradiction. Hence M is not supplemented.
Corollary 13.1: Let M be a hollow small quasi projective duo module. Then
M is not supplemented.
Proof: Follows from Proposition 2(a) and Proposition 13.
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Received: Received: August, 2009


Endomorphism Rings of Small Pseudo Projective Modules

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Endomorphism Rings of Small Pseudo Projective Modules

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