Internal. J. Math. & Math. Sci.
(1978)439-445
Vol.
439
A NOTE ON THE HOMOMORPHISM THEOREM FOR HEMIRINGS
D. M. OLSON
Department of Mathematics
Cameron University
73501
U.S.A.
Lawton, Oklahoma
(Recieved April II, 1978)
ABSTRACT.
The fundamental homomorphism theorem for rings is not generally
applicable in hemiring theory.
In this paper, we show that for the class of
N-homomorphism of hemirings the fundamental theorem is valid.
In addition,
the concept of N-homomorphism is used to prove that every hereditarily semisubtractive hemiring is of type (K).
KEY WORDS AND PHRASES. Hemirings homomorphism o f hemngs homomorphism
Theore,"N-homophism, Type (K) ideals, Hereditarily semubtaetive.
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
16A78.
440
i.
D.M. OLSON
INTRODUCTION.
It is well known that the analogue to the fundamental homomorphism theorem
However, in [I] Allen defined
is not necessarily true in general hemiring theory.
a class of maximal homomorphisms of hemirings for which the exact analogue could
In this article we extend his class to the class of N-homomorphisms
be proven.
for which the homomorphism theorem holds, and examine some properties of
N-homomorphisms.
Also, in [2] LaTorre defines the concepts of a hemiring being of type (H)
or type (K).
He gives results establishing certain classes of hemirings as being
all of type (H), but states that no general statement can be made concerning the
occurance of hemiring of type (K).
In section 4 we use the concept of an N-homo-
morphism together with the idea of a semisubtractive hemiring [4] to establish
that all hereditarily semisubtractive hemirings are of type (K).
In what follows we use the standard hemiring definitions and terminology
which may be found in [I] and
2.
[2].
N-HOMOMORPHISMS.
A hemiring homomorphism $ from S
DEFINITION i.
T,
maximal homomorphism if for every t e
for all
x e
-I
(t)
DEFINITION 2.
we have
x
+
A hemiring homomorphism
N-homomorphism if for every
t e
T,
.
there exists
ker $ c
t
+
ker
from
the collection
T
onto
c
t
is called a
-I
e
(t)
such that
[I]
S
omto
{x + ker $
T
is called an
x e $
-I
(t)}
contains no two sets which are disjoint.
It is easy to see that
if whenever
x
+ kI
follows.
y
(x)
+ k 2.
(y)
S
for some
/
T
will be an N-homomorphism if and only
x,y e S
we have
k
l,k 2
e ker $
such that
LaTorre [3] has also characterized maxiaml homomorphism as
441
HOMOMORPHISM THEOREM FOR HEMIRINGS
LEMMA 3.
A homomorphism
t e T
image of every
.
S /T is maximal if and only if the inverse
ker
is a coset of
With this lemma we can establish the following result.
THEOREM 4.
:’S + T
If
is an N-
is a maxml homomorphlsm then
homomo rphism.
Let
PROOF.
consider
x
+
ker #
x
and
x
that
c
+
+
/
kl,
ker
T
be a maximal homomorphlsm and let
+
y
and
ker
is a coset of
Let
S
ker
-I
say
y
and
y
+
ker #
#
(t)
.
+
c
for
k2,
+
c
.
x,y e
ker
then
x
+
k
(t).
Then
By Lemma i,
x,y e c
+ kI
y
2
and
t, x
Since
-i
y
t e T.
for
+ ker
k
l,k 2
Now
#
.
-I (t)
e ker
are arbitrary we see
is an N-homomorphism.
Since every ring homomorphlsm is maximal, these are also N-homomorphlsms.
In addition it is easy to verify that every natural may of a he.miring
a quotient hemiring
S/l
is an N-homomorphism.
S
onto
The following example shows that
the class of N-homomorphisms does not coincide with the class of maximal homo-
morphisms.
EXAMPLE 5.
Let
S
{0,1,2,3,4}
be the hemiring with zero multiplication
and addition defined by the following table.
Let
T
be the subhemiring
{0,i}
0
1
2
3
4
0
1
2
3
4
1
1
4
4
4
2
4
4
4
4
3
4
4
4
4
4
4
4
4
4
of
S
Define
S
/
T
by
0,i
/
0
D.M. OLSON
442
2,3,4-
and
-I(I)
Now
Thus
are disjoint.
of
ker
{2,3,4}
so
2
and no two of
+
3
+
and 4
ker #
-I
However,
is an N-homomorhpism.
#
,
ker
{0}.
ker
is a hemiring homomorphism with
then
1
(I)
+
ker
is not a coset
is not a maximal homomorphism.
THE FUNDAMENTAL HOMOMORPHISM THEOREM
3.
For the class of N-homorphism we can establish analogues to certain desirable
results from ring theory.
LEMMA 6.
then
T
with
l,k2
e ker
onto
(0),
ker
is an isomorphism.
PROOF.
x
S
is an N-homomorphism from
If
+ kI
Suppose
+ k2.
y
(x)
(y).
k
Then there exist
(0), k I
But since ker
0.
k
2
Thus
x
such that
y
and
is
an isomorphism.
THEOREM 7.
PROOF.
If
Define
equivalence class of
usual
#(s)
:
S/ker
s
in
/
S
T
by
([s])
such that
+ kI
t
(s), where
ker
([s])
If
then
S/ker #
[s]
is the
of
+ k2.
But then by definition
[s]
S.
([t]),
is an N-homomorphism, there must exist
and, since
s
T
onto
determined by the ideal
is a well defined onto homomorphism.
(t)
S
is an N-homomorphism from
It]
T.
Then as
then
kl,k2,
and
e ker
is an
isomorphism.
It is clear that the class of N-homomorphism is the largest class for which
as defined in the proof of Theorem 7, will be an isomorphism.
the mapping
THEOREM 8.
ideal of
T,
PROOF.
If
then
Define
is an N-homomorphism from
s/
-I
(K)
S
onto
T
and
K
is an
T/K.
S + T/K
by
(s) --[ # (s)]
Then one can quickly check
HOMOMORPHISM THEOREM FOR HEMIRINGS
to see that
S
is a homomorphism from
then
[(s)]
kl,k2
k.
Then
(s +
[(t)]
which implies that
l
Choose
and
I
(t +
i
12
2 )"
from
T/K.
onto
(t)
for some
respectively.
is an N-homomorphism, there exist
Since
-I(K) such that s + (i + Zl
Ii + Zl’ 2 + z2 -I(K)’ they are both in ker
c_
ker
Zl,Z2
(t) + k2
-l(k2)
and
(s)
Now if
(s) + k I
-l(kl)
443
t
+
(2 + z2 )"
and thus
But since
is an N-homo-
morphism.
-l(K)*
Finally
But if
s
+
k
{s
ker
[(s)]
@(s)
S
0}
{s
E
(s)
S
-I (K). Thus
K * then (s) + (k)
K for some k
(s)
-I
-I
-i
(K) so s
(K) * This gives us that ker
(K) *
,
S/-I(K) S/-I(K)
the preceeding theorem
E
K*}.
so by
T/K.
HEMIRINGS OF TYPE K
4.
A hemiring
DEFINITION 9.
k-ideal of
S
n
and
S
/
S/I
S
(K)
is of type
I
provided that if
is a
n preserves
is the natural homomorphism, then
k-ideals. [2]
A hemiring S
DEFINITION i0.
pair of elements
a
+
b
x
+
b
or
and
a
x
a
S
in
S
at least one of the equations
is solvable in
A hemiring
DEFINITION 11.
ideal of
b
is said to be semisubtractive if for every
S
is
S.
[4]
hereditarily semisubtractive if each
is semisubtractive as a hemiring.
Clearly every ring is hereditarily semisubtractive and also the hemiring of
non-negative integers under the operations of
a
b
min{a,b}.
If
+
b
max{a,b}
and
In view of the following lemma any semisubtractive hemiring
whose ideals are all
LEMMA 12.
a
k-ideals
S
is also hereditarily semisubtractive.
is a semisubtractive hemiring and
K
is a
k-ideal of
444
S
D.M. OLSON
K
then
is semisubtractive.
Let
PROOF.
a,b e K.
for which either
a
+
s
+
a
to be
s
requires
THEOREM 13.
K.
S
K
of
s
under
some
k
that
x
of
S
+
For the sake of argument say
T
K
+
z
2
k
t e K
+ zI
#(k I + z 2 + t)
2
+
k
2
(x + k 1)
+ z 2.
+ kI + z I + t
we have x + k +
I
x
+
get
x
+ zI +
K
K
z
z
+
2
is of type
K-- (K).
Then
#(K)
and so
We shall show that
x e K
If
ker
+
ker #
z
I
+ kI +
x
for the image
s
then
x
+ kI
k
e ker
Zl,Z2
for
2
such
is semisubtractive as an ideal
such that
+
2
kl+
z
t
2
or
t.
+ zI
t
k
2
+
z
+
2
t
which implies
(x + z I + z 2)
(x)
+ kI + z I +
Now t e K + ker
K.
COROLLARY 14.
+
k
Then
t.
k-ldeals.
so there exist
t e K
x
is an N-homo-
K.
2
3
K
such that
+ k I.
k + (z +
3
I
k
is a k-ideal this
We use the notation
# (k 2)
Since
ker
(K)
e
K
and since
and
k-ideal.
In the xecond case we get
have
S
k-ideal of
In the first case we see that
x
a e K
preserves
we have either there exists
that
a.
is indeed semisubtractive.
then
is a
Then
+ kI + z I
there exists
s
whenever it is convenient.
#
e K.
l,k 2
there exists an element
with
K
s e S
a,b e S,
is hereditarily semisubtractive and
be a
K C_ K and thus that
b
s e K
Thus
onto
Let
PROOF.
+
a
If
S
morphism from
or,
b
s
But then
b.
Since
If
S
x
3)
z
x
K
+ k I.
(x + z I +
k
is a
2
k
t
so
3)
k-ideal of
+ z 2 + t so
k + z
t
and as
3
3
K
Since
z
2
is a
(K)
e
T
K.
k-ideal of
a result
S
we
In any case we
as desired.
is an hereditarily semisubtractive hemiring then
S
(K).
PROOF.
homomorphism.
If
I
is a
k-ideal of
By Theorem 13
n
S
the natural map
preserves
n
S
/
k-ideals which makes
S/I
S
is an N-
a hemiring
HOMOMORPHISM THEOREM FOR HEMIRINGS
445
of type (K).
REFERENCES
i.
Allen, Paul J., "A Fundamental Theorem of Homomorphism for Semirings," Proc.
Amer. Math. Soc. 21 (1969) 412-416.
2.
LaTorre, D. R., "On h-ideals and k-ideals in Hemirimgs,"
Debrecen 12 (1965) 219-226.
3.
LaTorre, D. R., "A Note on Quotient Semirings," Proc. Amer. Math. Soc.
(1970) 463-465.
4.
Mosher, James R., "Generalized Quotients of Hemirings," Compositio Math.
22 (1970) 275-281.
Publ. Math.
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