A FUNDAMENTAL THEOREM OF HOMOMORPHISMS

FOR SEMIRINGS
PAUL J. ALLEN
1. Introduction. When studying ideal theory in semirings, it is
natural to consider the quotient structure of a semiring modulo an
ideal. If 7 is an ideal in a semiring R, the collection {x+l}xeit o sets
x + I={x+i\iEl} need not be a partition of R. Faced with this
problem, S. Bourne [l], D. R. La Torre [3] and M. Henriksen [2]
used equivalence relations to determine cosets relative to an ideal.
La Torre successfully established analogues of several well-known
isomorphism theorems for rings. However, the methods that Bourne
and La Torre used to construct quotient structures proved to be un-
successful when trying to obtain an exact analogue of the Funda-
mental Theorem of Homomorphisms.
In this paper, the notion of a -ideal will be defined and a construc-
tion process will be presented by which one can build the quotient
structure of a semiring modulo a Q-ideal. Maximal homomorphisms
will be defined and examples of such homomorphisms will be given.
Using these notions, the Fundamental Theorem of Homomorphisms
will be generalized to include a large class of semirings.
2. Fundamentals. There are many different definitions of a semi-
ring appearing in the literature. Throughout this paper, a semiring
will be defined as follows:
Definition 1. A set R together with two associative binary opera-
tions called addition and multiplication (denoted by + and , respec-
tively) will be called a semiring provided:
(i) addition is a commutative operation,
(ii) there exist OE^? such that x+0=x and a;0 = 0;*; = 0 for each
xER, and
(iii) multiplication distributes over addition both from the left and
from the right.
Definition 2. A subset 7 of a semiring R will be called an ideal if
a, bEI and rER implies a+bEL raE7 and arEI-
Definition 3. A mapping v from the semiring R into the semiring
R' will be called a homomorphism if (a+b)r)=ar)+bn and (ab)n = ar\bn
for each a, bER- An isomorphism is a one-to-one homomorphism. The
Presented to the Society, January 25, 1968 under the title Quotient structure of a
semiring; received by the editors July 1, 1968.
412
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A THEOREM OF HOMOMORPHISMS FOR SEMIRINGS 413
semirings R and P' will be called isomorphic (denoted by R==R') if
there exists an isomorphism from R onto R'.
3. Quotient structures. The notion of a Q-ideal will now be defined
and a construction process will be presented by which one can build
the quotient structure of a semiring with respect to a Q-ideal.
Definition 4. An ideal / in the semiring P will be called a Q-ideal
if there exists a subset Q of R satisfying the following conditions:
(1) {<7 + -Me<3 's a partition of P; and
(2) if qi, q,QQ such that qi9*qi, then (qi + I)(~\(qi + I)=0.
It is clear that every ring ideal / in a ring P is a C/-ideal. The fol-
lowing examples will show that Q-ideals do occur in semirings that
are not rings.
Example 5. Let R be a nonempty, well ordered set and define
a+6 = max (a, b) and a = min (a, b) for each a, bQR. R together
with the two defined operations forms a semiring. If rQR, then the
set P= {xGP|x^r} is an ideal in P. It is clear from the definition of
addition in R that 0+P = P and x+P= {x} for each x>r. Thus, Ir
isa Q-ideal when Q = {o} \j{xQR\x>r}.
Example 6. Let Z+ denote the semiring of nonnegative
integers with the usual operations of addition and multiplication.
If mQZ+{0}, the ideal (m)= {nm\nQZ+} is a Q-ideal when
Q = {0, 1, , m l}. If w = 0, the ideal (m) is a Q-ideal when
Q = Z+. A simple argument will show the ideal Z+ {l} can not be
a Q-ideal.
Lemma 7. Let I be a Q-ideal in the semiring R. If xQR, then there
exists a unique qQQ such that x+IQq+I.
Proof. Let xQR. Since {<?+/} qeq is a partition of R, there exists
qQQ such that xQq+I. If yGx+P there exists iiQI such that
y=x+4i. Since xQq+I, there exists42Gesuch that x = a+42. Clearly,
y=x+4i=(a+42)+4'i = a + (42+4i)Ga+P Thus, x+IQq+I. The
uniqueness is an immediate result of part (2) of Definition 4.
Let / be a Q-ideal n the semiring P. In view of the above result,
one can define the binary operations g and Oq on {q+l}teQ as
follows:
(1) (ai-f-7) q (qi+I) =qz+I where q3 is the unique element in Q
such that qi+qi+IQq3+I; and
(2) (qi+I)O (qi+I) =q+I where q3 is the unique element in Q
such that qiqi+IQqs+I. The elements qi+I and qi+I in {a+/} qeQ
will be called equal (denoted by qi+I = qi+I) if and only if qi = qi.
Theorem 8. // I is a Q-ideal in the semiring R, then
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414 P. J. ALLEN [May
({? + 7}a60, g, Oq)
is a semiring.
Proof. It is an easy matter to show that o and O o are asso-
ciative operations, ffig is a commutative operation, and Oq dis-
tributes over g both from the left and from the right. Define
<t>: R>{q+l} qeQ by x<p = q + I where g is the unique element in such
that ic+7Ea+7. It can be shown that 0 is a homomorphism from
the semigroup (R, +) onto the semigroup ({q + l} qEQ, Oq) and <p
is a homomorphism from the semigroup (R, ) onto the semigroup
({q + l} qEQ, Oq). Since 0 is the identity in (R, +), it follows that
0<p = q*+lis the identity in ({q + l}qeQ, Oq)- Let qEQ and let xER
such that x<j> = q + I. Since *0=0;t = 0, it is clear that g*+7 = O0
= (Qx)<t> = 0<f>x<p = (q* +1) Oq (g-f-7) and g*+7 = O0 = (xO)<p = x<t>Od>
= (q + I) Oq (q*+I). Thus, the element g*+7 satisfies condition (ii)
in Definition 1.
Theorem 9. Let I be an ideal in the semiring R. If Qi and Q2 are sub-
sets of R such that I is both a Qi-ideal and a Q2-ideal, then
({q + Keg,, ,, Ofli) = ({? + l},Qi, ,, Oq,)-
Proof. Define r:{q+l} qQi-^{q + l} qeQl as follows: If qiEQi,
then (i + 7)77=g2 + 7 where g2 is the unique element in Q2 such that
gi-f-7C2 + 7. It can be shown that r is an isomorphism from the
semiring ([g+7}aeQl, Ql, O0l) onto the semiring
({q + l]teQ 2, Oo2).
If 7 is an ideal in the semiring R, then it is possible that 7 can be
considered to be a Q-ideal with respect to many different subsets Q o
R. However, the preceding theorem implies that the structure
({q + l}qeQ, q, Oq) is "essentially independent" of the choice of Q.
Thus, if 7 is a -ideal in R the semiring ({g+7} 8gq, Oq, Oq) will be
denoted by R/I or (R/I, , O).
4. Maximal homomorphisms.
Definition 10. A homomorphism rj from the semiring R onto the
semiring R' is said to be maximal if for each aER' there exists
CoE^Ka}) such that x+ker(-q)Eca+ker(t]) for each xEr-1({a}),
where ker(r;) = {xE7^|xt7=0}.
If t; is a homomorphism from a ring R onto a ring R', it is well
known that *+ker(77) =y + ker(i7) for each x, yEy~1({a}), aER'-
Thus, any ring homomorphism is a maximal homomorphism. Unfor-
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i969] A theorem of homomorphisms for semirings 415
tunately, the following example shows that semiring homomorphisms
need not be maximal.
Example 11. The set P of nonnegative integers is well ordered
under the usual ordering of the integers. Thus, P can be considered
to be a semiring as described in Example 5. Clearly, P'= {0, 1} is a
subsemiring of P. Define -n: R*R' by x?7 = 0 if x = 5 and xr = if
x>5. It can be shown that r) is a homomorphism from P onto P'.
Since ker(-q)= {xQR\x^5}, it is clear that y + ker(r]) = {y}, for
each yQr]~1( {1} ). Thus, there does not exist CiQr~1({1} ) such that
y-r-ker(7?)Cci + ker(77), for each yGT'({l})-
The following examples will show there exist maximal homo-
morphisms other than ring homomorphisms.
Example 12. The set P of nonnegative real numbers with the
usual ordering forms a semiring as described in Example 5. Let
S'= {n/2QR\n = \, 2, 3, } and 5= \xQR\ Ogxgl/4} US'. It
is clear that 5 and S' are subsemirings of P. If r:S>S' is defined by
xr = 0, ixQS and 0 = x g 1/4,
= w/2, if x G 5 and x = /2,
then it can be shown that r\ is a maximal homomorphism.
Example 13. Let Z+ denote the semiring of nonnegative integers
described in Example 6, and let Z/(m) denote the ring of integers
modulo (m) where m>0. If xQZ+, the division algorithm implies
there exist unique integers? and r such that x = qm+r where 0^r<m.
Define rj:Z+-^Z/(m) by xr = r + (m) where r is the unique integer
described above. 77 is a maximal homomorphism from Z+ onto
Z/(m).
5. A fundamental theorem of homomorphisms. Whenever 77 is a
maximal homomorphism, ca will denote an element in ^""'(ja}) such
that xQr)~1({a}) implies x-r-ker(7)Cca + ker(t). With the aid of this
notation and the following lemmas, an analogue of the Fundamental
Theorem of Homomorphisms can be obtained.
Lemma 14. Let r be a homomorphism from the semiring R onto
the semiring R'. If r\ is maximal, then ker(rj) 45 a Q-ideal, where
Q JCa/oS'-
Proof. It is clear that Uae'(ca+ker(ji))=P. Let ca and c& be
distinct elements in Q; i.e., a9*b. Assume (ca + ker(r))r\(ct + ker(r))
9*0. Thus, there exist k, k'Q ker(tj) such that ca+k=ci,+k'. Thus,
a = car]+kri = (ca+k)ri = (ci>+k')r)=Cb't]+k'ri = b, a contradiction. It
now follows that ker(??) is a Q-ideal.
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416 P. J. ALLEN
Lemma 15. Let R, R', n and Q be as stated in Lemma 14, and let ca,
cb and cc be elements in Q.
(1) 7/c0+C6 + ker(77)Ccc+ker(7?), then a+b=c.
(2) If caCb+ker(rj)Ecc+ker(ri), then ab=c.
Proof. Since ca+C6Eca+Ct + ker(7j)CCc + ker(r/), there exists
kEker(v) such that ca+C6 = cc+&. Thus, a+b=caT]+cbr] = (ca+cb)n
= (cc+k)r) = ccr)+kr)=c. A similar argument shows (2) is true.
Theorem 16. If n is a maximal homomorphism from the semiring R
onto the semiring R', then R/ker(rj)^R'.
Proof. Define ij:R/ker(r])*R' by (c0+ker(;))^ = a, for each
caEQ- It is clear that rj is a one-to-one function from i?/ker(??) onto
R'. It will be shown that rj is an isomorphism and the theorem will
follow. From the definition of addition in i?/ker(i;), it follows that
[(ca + ker(7;))(c6 + ker(77))]jj= [cc+kei(?;)]^=c, where cc is the
unique element in Q such that c + c&+ker (77) Ecc+ker (77). In view of
Lemma 15, it is clear that
(ca + ker(7?))7j + (cb + kev(rj))rj
= a + b = c = [(ca + ker(v)) (cb + ker(ij))]ij.
The definition of multiplication in R/ker(r) implies
[(ca + ker(j)) O (cb + ker(r]))]rj = [cc + ker(i7)]ij = c,
where cc is the unique element in Q such that cacb+ker(77) Ecc+ker(?7).
In view of Lemma 15, it is clear that (c0 + ker(?7))^(c6 + ker(77))j=a&
= c= [(c0+ker(?7))0(c6 + ker(T7))]^.
References
1. S. Bourne, On the homomorphism theorem for semirings, Proc. Nat. Acad. Sei.
U.S.A. 38 (1952), 118-119.
2. M. Henriksen, h-ideals in semirings, Notices Amer. Math. Soc. 5 (1958),
321.
3. D. R. La Torre, On h-ideals and k-ideals in hemirings, Publ. Math. Debrecen
12 (1965), 219-226.
University of Alabama
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