MINISTRY OF EDUCATION AND TRAINING
VINH UNIVERSITY
TRN GIANG NAM
MORITA EQUIVALENCE FOR SEMIRINGS AND
CHARACTERIZE SOME CLASSES OF SEMIRINGS
ABSTRACT OF THE DOCTORAL THESIS
Speciality: Algebra and theory of numbers
Code: 62.46.05.01
Supervisors
1. ASSOC. PROF. SCI. DR. NGUYEN XUAN TUYEN
2. ASSOC. PROF. DR. NGO SY TUNG
VINH - 2011
1
Introduction
Semiring is introduced by Vandiver in 1934, generalize the notion of non-
commutative rings in the sense that negative elements don’t have exist. In this
thesis, semirings were assumed to have both additive identity and multiplicative
identiy.
Nowadays one may clearly notice a growing interest in developing the alge-
braic theory of semirings and their numerous connections with, and applications
in, different branches of mathematics, computer science, quantum physics, and
many other areas of science (see, for example, the recently published Glazek
(2002)). As algebraic objects, semirings certainly are the most natural gen-
eralization of such (at first glance different) algebraic systems as rings and
bounded distributive lattices. As is well known, structure theories for alge-
bras of classes/varieties of algebras constitute an important “classical” area of
the sustained interest in algebraic research. In such theories, so-called simple
algebras — algebras possessing only the identity and universal congruences —
play a very important role of “building blocks.” In contrast to the varieties of
groups and rings, research on simple semirings has been just recently started
and, therefore, not much on the subject is known (for some recent results on

In Chapter 4, we describe all additively idempotent semisimple semirings
over that the notions of either projectiveness and flatness, or flatness and mono-
flatness for semimodules coincide. Also, we characterize semisimple semirings
by projective and injective semimodules.
3
Chapter 1
MORITA EQUIVALENCE
1.1 Background
In this section, we recall notions and facts on semirings and semimodules over
them that can be found in Golan (1999). Also, we recall the notion of tensor
product for semimodules in sense of Y. Katsov (1997).
From now on, let M
R
and
R
M be the categories of right and left semimodules,
respectively, over a semiring R.
1.2 Progenerators
In this section, we characterize, and describe progenerators — finitely
generated projective generators — of semimodule categories.
Now let S =
R
M(P, P ) := End (
R
P ) be a semiring of all endomorphisms
of a left R-semimodule
R
P ∈ |
R
M|. Then, considering endomorphisms of

classical case of modules over rings, one can easily obtain the following.
Lemma 1.2.2. The assignments (p, q) −→ pq and (q, p) −→ qp define
the (R, R)-homomorphism α : P ⊗
S
Q −→ R and the (S, S)-homomorphism
4
β : Q ⊗
R
P −→ S, respectively.
Our next observation provides a characterization of finitely generated pro-
jective left R-semimodules
R
P ∈ |
R
M| by means of the homomorphism
β : Q ⊗
R
P = P
∗
⊗
R
P −→ S = End (
R
P ) from Lemma 1.2.2 above.
Proposition 1.2.4. A left R-semimodules
R
P ∈ |
R
M| is finitely generated and
projective iff β : Q ⊗

Q −→ R from Lemma 1.2.2. above.
Proposition 1.2.8. A finitely generated left semimodule
R
P ∈ |
R
M| is a
generator for
R
M iff the (R, R)-homomorphism α : P ⊗
S
Q −→ R is a
surjection. Moreover, if α is a surjection, then it is an isomorphism.
Combining the concepts of ‘generator’ and ‘finitely generated projective
semimodule’, we come up with a new concept of ‘progenerator’, namely: A
left semimodule
R
P ∈ |
R
M| is said to be a progenerator for the category of left
semimodules
R
M if it is a finitely generated projective generator. Then, from
Propositions 1.2.4 and 1.2.8, one obtains the following important result.
Theorem 1.2.9. A left R-semimodules
R
P ∈ |
R
M| is a progenerator iff the
homomorphisms α : P ⊗
S

−→ M
S
the following statements are
equivalent:
(i) F has a right adjoint;
(ii) F is right continuos and preserves coproducts (direct sums);
(iii) There exists unique up to natural isomorphism a R-S-bisemimodules
P ∈ |
R
M
S
| such that the functors − ⊗
R
P : M
R
−→ M
S
and F are naturally
isomorphic, i.e., F
∼
=
− ⊗
R
P .
Using Theorem 1.3.5, one gets the following important and intriguing con-
sequence of the Morita equivalence between two semirings R and S is the fact
that reasonable important corresponding categories of semimodules over these
semirings are equivalent, as categories, as well.
Theorem 1.3.12. For semirings R and S the following conditions are equiva-
lent:

, and universal, R
2
,
congruences are the only congruences on R; and R is ideal-simple if 0 and R
are its only ideals. A semiring R is said to be simple if it is simultaneously
congruence- and ideal-simple.
Congruence-simple and ideal-simple semirings have been studied by C. Mon-
ico (2004), J. Zumbragel (2008), J. Jeˇzek - T. Kepka - M. Mar´oti (2009), Bourne-
Zassenhaus (1957), O. Steinfeld - R. Wiegandt (1967), Stone (1977), Weinert
(1984),
A surjective homomorphism of semirings f : R −→ S is called strongly
semiisomorphic if Ker(f) := f
−1
(0) = {0} and f(I)  S for any proper ideal I
of R.
A semiring R is called additively idempotent if the monoid (R, +) is additively
idempotent. Our next proposition illustrates that ideal-simple semirings can be
studied by additively idempotent simple semirings.
Proposition 2.1.6. A semirng R is ideal-simple iff R is a simple ring, or
there exists a strong semiisomorphism from R onto an additively idempotent,
simple semiring S.
8
In the light of Proposition 2.1.6, it is natural to bring up that ideal-
simple (simple) semirings can be understood by simpleness of subsemirings of
endomorphism semirings of idempotent commutative monoids. Our next result
describes simple endomorphism semirings of idempotent commutative monoids.
Theorem 1.2.9. The following conditions for the endomorphism hemiring
End(M) of an idempotent commutative monoid (M, +, 0) are equivalent:
(i) End(M) is simple;
(ii) End(M) is ideal-simple;

i
(D) (i ≥ 0) of square matrices of order 2
i
over D. We shall regard R
i
as a subsemiring of R
i+1
by identifying a 2
i
× 2
i
matrix M with the 2
i+1
× 2
i+1
9
matrix

M 0
0 M

. In this way, we have a chain of semirings
R
0
⊆ R
1
⊆ R
2
⊆ . . . ,
where R

D = End (
R
I) (viewed as a semiring of right operators on I). Then
(i) The natural map f : R −→ End (I
D
) is a semiring isomorphism;
(ii) I is a generator in the category of semimodule
R
M, and a finitely
generated projective right D− semimodule;
(iii) There exists a natural number n and an idempotent e in matrix semiring
M
n
(D) such that R
∼
=
eM
n
(D)e;
(iv) D is simple iff I is a finitely generated projective left R-semimodule.
As a corollary of Theorem 2.3.1, we obtain the description of all simple
semirings having projective minimal left (right) ideals.
Theorem 2.3.2. For a semiring R, the followings are equivalent:
(i) R is a simple semiring containing a projective minimal left ideal;
(ii) R is a simple semiring containing a projective minimal right ideal;
10
(iii) R is either isomorphic to a matrix semiring M
n
(F ) for some division
ring F and n ≥ 1 , or ismorphic to E

total, (R, ≤) forms a chain, and the semiring R is called an additively idempotent
chain semiring or, in short, aic-semiring, by Takahashi-Wang (1993).
Let G be a totally-ordered multiplicative group and R := G ∪ {0}. Then,
extending the order on G to R by setting 0 ≤ g for any g ∈ G, and defining
0g = g0 = 0 for all g ∈ G, one has that (R, max, ·) is a division aic-semiring,
and called ”max–plus” semiring. Kt qu di y m t cu trc ca na vnh Artin tri (phi)
xch khng c ian khng tm thng v na vnh Artin tri (phi) xch n.
Our following result describes all ideal-simple and simple artinian aic-
semirings.
Theorem 3.1.4. (i) A left (right) artinian aic-semiring R is ideal-simple iff it
is a division aic-semiring.
(ii) A left (right) artinian aic-semiring R is simple iff R
∼
=
B.
A semiring R is lattice-ordered if and only if there also exists the lattice
12
structure (R, ∨, ∧) on R such that a + b = a ∨ b and ab ≤ a ∧ b for all a, b ∈ R
with respect to the partial order naturally induced by the lattice operations.
Our next observation describes all congruence-simple lattice-orderd semirings.
Theorem 3.1.5. For a lattice-ordered semiring R the following statements are
equivalent:
(i) R is congruence-simple;
(ii) R is simple;
(iii) R
∼
=
B.
3.2 Subtravtive semisimple semirings
A left ideal I of a semiring R is said to be subtractive ideal if for all a, b ∈ R

1
) × · · · × M
n
r
(R
r
), where D
1
, . . . , D
n
are
proper division semirings, R
1
, . . . , R
r
are division rings, n ≥ 0, r ≥ 0, positive
integers n
1
, . . . , n
r
are greater 1.
As an application of Theorem 3.2.6, the following result describes all left
(right) subtractive artinian ideal-simple semirings.
Theorem 3.2.7. A left (right) subtractive, left (right) artinian semiring R is
ideal-simple iff (1) R
∼
=
M
n
(R

is defined and
satisfies the following conditions:
(1) Σ
i∈∅
x
i
= 0, Σ
i∈{1}
x
i
= x
1
, Σ
i∈{1,2}
x
i
= x
1
+ x
2
;
(2) If I = ∪
j∈J
I
j
is a partition, then Σ
j∈J
(Σ
i∈I
j

) = ∨
i∈I
f(m
i
), for every set
{m
i
| i ∈ I} ⊆ M. Then, CEnd(M) can be made a complete semiring with
the infinite summation to be defined by (Σ
i∈I
f
i
)(m) = ∨
i∈I
f
i
(m) for every set
{f
i
| i ∈ I} ⊆ CEnd(M) and m ∈ M.
14
In next theorem, one gives a complete description of congruence-simple
complete semirings.
Theorem 3.3.6. For a complete semiring R, the following conditions are
equivalent:
(i) R is congruence-simple;
(ii) R is isomorphic to a complete subhemiring S ⊆ CEnd
B
(M) of the
complete endomorphism hemiring CEnd

15
semisimple semirings (Theorem 3.2.6). Then, apply this result, we describe left
(right) subtractive artinian ideal-simple semirings (Theorem 3.2.7), as well as
left (right) subtractive artinian congruence-simple semirings. Finally, describing
all simple semirings with an infinite element (Theorem 3.3.9), and provide a
characterization of ideal-simple semirings with an elements (Theorem 3.3.10).
16
Chapter 4
Homological characterization of
semirings
4.1 Semisimple and subtractive semirings
A homomorphism f : A
R
−→ B
R
is k-regular if the folowing statement
∀a, a

∈ A(f(a) = f(a

) ⇐⇒ ∃k, k

∈ Ker(f) : a + k = a

+ k

)
is true.
A semimodule U
R

R-homomorphism f : M −→ U and the natural embedding i : K  M; and a
semimodule U
R
∈ |M
R
| is k-injective iff it is M-k-injective for any semimodule
M
R
∈ |M
R
|.
A semimodule G ∈ |
R
M| is said to be mono-flat if µ ⊗ 1
G
: F
1
⊗ G  F ⊗ G
is a monomorphism in M for any monomorphism of right R-semimodules
µ : F
1
 F , and G ∈ |
R
M| is flat if the functor − ⊗
R
G : M
R
−→ M
17
preserves finite limits.

ules coincide are just perfect rings, and expressed a conjecture that the same
situation would take place in the entire class of additively regular semirings.
Our following result confirms this conjecture for additively regular semisimple
semirings.
Theorem 4.2.1. The following conditions on an additively regular semisimple
semiring R are equivalent:
(1) The concepts of the ‘projectiveness’ and ‘flatness’ for left R-semimodules
coincide;
(ii) R is a semisimple ring.
In 1978, Bulman-Fleming and McDonwell showed that a left B-semodule A
is flat iff A is mono-flat, iff A is a distributive semilattice. In 1986, E. B. Katsov
extended this result to finite Boolean algrbras. Recently, in 2004, Y. Katsov
extended the above result to arbitrary Boolean algebras. chng minh c rng khng
nh trn vn cn ng i vi cc na mun trn i s Boole bt k. Also, he expressed the following
problem: “Describe the class of semirings over which the notions of flatness and
mono-flatness for semidules coincide?”. Our next result solves this problem in
the class of additively regular semisimple semirings.
Theorem 4.2.4. The following conditions on an additively regular semisimple
semiring R are equivalent:
(1) The concepts of the ‘mono-flatness’ and ‘flatness’ for right (left) R-
semimodules coincide;
(2) R
∼
=
M
n
1
(D
1
)× · · · ×M

notions of either projectiveness and flatness, or flatness and mono-flatness for
semimodules coincide. Also, characterize semisimple semirings by projective
and injective semimodules.
20
The list of main publications
1. Y. Katsov, T. G. Nam, N. X. Tuyen (2009), On subtractive semisimple
semirings, Algebra Colloquium, 16, 415-426.
2. Y. Katsov, T. G. Nam, N. X. Tuyen (2011), More on Subtractive Semirings:
Simpleness, Perfectness and Related Problems, Communications in Algebra
(accepted).
3. Y. Katsov and T. G. Nam (2011), Morita Equivalence and Homological
Characterization of Semirings, Journal of Algebra and Its Applications, 10,
449–473.
4. Y. Katsov, T. G. Nam and J. Zumbragel, On Simpleness of Semirings and
Complete Semirings (submitted).