On the Weak Grothendieck Group of a Morphic Ring and its Representations

The K-theoretical aspect of the commutative mophic rings is established using the arithmetical properties of the morphic rings in order to obtain a ring of all Smith normal forms of matrices over the morphic ring. The internal structure and basic properties of such rings are discussed as well as their presentations by the Witt vectors. In a case of a commutative von Neumann regular rings the famous Grothendieck group K0(R) obtains the alternative description.


Introduction
In [10] it is proved that for any element a of a commutative morphic ring R there is an element b ∈ R such that the ideals aR, bR coincides with the annihilators Ann(b), Ann(a). Therefore, one can make a partition on the pairs of the set of all principal ideals such that any element of such of this partition is uniquely determined by some pair of principal ideals. So, the structure of principal ideals determines all the properties of the morphic rings.
We will construct an analogue of the Grothendieck group K 0 (R) of a ring R using the principal ideals instead of the finitely generated projective R-modules. Such abelian group, that is denoted as K ′ 0 (R) and is called a weak Grothendieck group of a ring R, becomes a ring if we define a product of two elements of this group using the tensor product of principal ideals. Moreover, the elements of such ring can be interpreted as classes of equivalence of the Smith normal forms of the matrices over a ring R. In order to obtain a convenient way for the multiplication and the addition for the elements of K ′ 0 (R) we will establish the connection with the subring W ′ (G(R))of the ring of Witt vectors over some ring. As a direct consequence it will be proved that the functors K ′ 0 and W ′ G are naturally equivalent.
In a case of a commutative von Neumann regular ring the structure of K ′ 0 (R) becomes simpler and, as a consequence, K ′ 0 (R) and the usual Grothendieck group K 0 (R) coincide.
The main motivation of these investigations is that in [11]it is proved that a commutative Bezout domain is an elementary divisor ring if and only if any quotient ring R/aR is so, where a is an arbitrary nonzero element of R. Since any finite homomorphic image of a commutative Bezout domain R is a morphic ring [14] then the studies of the ring K ′ 0 (R/aR) become related to the famous elementary divisor ring problem [8].

Preliminaries
All the rings considered in the article are supposed to be commutative with the nonzero identity element. By the semiring R we mean a set that satisfies all axioms of a ring except the obligatory existence of additive inverses. The subsemiring S of a semiring R is defined as a subset that is a semiring itself with respect to the addition and multiplication induced from R. The simplest examples of semirings are nonnegative integers, set of all subsets of some finite set with respect to the union and intersection operations. For more information we refer to [3]. Let U (R) be a set of all invertible elements of a ring R. By the Jacobson radical J(R) of a ring R we mean the set J(R) = {x ∈ R|∀a ∈ R : 1 − ax ∈ U (R)}, and the nilradical N il(R) is defined as an ideal of all nilpotent elements of a ring R. A ring R is called a reduced ring if Suppose that A is a subset in a ring R. A set Ann(A) = {x|Ax = 0} is called an annihilator of a set A. If an element a ∈ R is a divisor of an element b ∈ R then we will write a|b.
We start with recalling of some definitions and facts that we will need below in our proofs. Definition 1. 1) If any finitely generated ideal of a ring R is principal then a ring R is said to be a Bezout ring.
2) We say that a rectangular matrix A over a ring R admits canonical diagonal reduction if there are two invertible matrices P, Q of the appropriate sizes such that the matrix is a diagonal matrix with an additional condition: for all indices we have the inclusion of the ideals d i+1 R ⊆ d i R.
3) If every row matrix (a, b) (column matrix (a, b) T ) admits canonical diagonal reduction then we say that R is a right (left) Hermite ring. 4) If every matrix over a ring R admits canonical diagonal reduction then R is said to be an elementary divisor ring.
It is not odd to say that any elementary divisor ring is an Hermite one, and the Hermite rings are Bezout rings. An examples of Bezout rings are all principal ideal rings, the rings of real-valued functions over the topological F-space, all polynomial rings over von Neumann regular rings. As more narrow class elementary divisor rings include the ring of entire functions, the ring of algebraic integers, all adequate rings, the Bezout rings of stable range 1. For more information see [13].
Definition 2. We say that a ring R has the stable range 1 if for any elements a, b ∈ R the equality aR + bR = R implies that there is some x ∈ R such that (a + bx)R = R [1].
Remark that the last definition is left-right symmetric due to [12].

Definition 3.
A ring R is called a morphic ring if for any a ∈ R there is an R-module isomorphism R/aR ∼ = Ann(a) [10].
Here is a Nicholson's criterion for a morphic ring.
Theorem [10]. The following statements are equivalent for a ring R: a) R is a morphic ring; b) For any a ∈ R one can find b ∈ R such that Ann(a) = bR, Ann(b) = aR; c) For any a ∈ R one can find b ∈ R such that Ann(a) = bR, Ann(b) ∼ = aR . Also in [10] it is proved every the commutative von Neumann regular ring is a morphic one. In addition it is useful to mention that a pair (a, b) of elements of a ring R in the previous theorem is called a morphic pair and this fact will be denoted as aR ∼ bR, since any morphic pair is determined by the pair of some principal ideals, but not elements. Definition 4. An element a of a ring R is said to be a von Neumann regular element if there is some b ∈ R such that aba = a. If all elements of a ring R are von Neumann regular then R is called a von Neumann regular ring.
The finite direct sums of fields, boolean rings, and rank rings are examples of von Neumann regular rings. For more details see [4].

Weak Grothendieck group
Let R be a commutative morphic ring. We will try to construct an analogue K ′ 0 (R) of the Grothendieck group K 0 (R) considering the isomorphism classes of the finite direct sums of the principal ideals of R as the basic objects and using some ideas from [9].
Let ∆(R) = {a 1 R ⊕ ... ⊕ anR|a 1 , ..., an ∈ R} be a set of all finite direct sums of the principal ideals of R. Then we consider a relation "∼" on the set ∆(R) defined as Then let F (R) be a free abelian group generated by the set ∆(R)/ ∼. Since every element of ∆(R)/ ∼ is in the one-to-one correspondence with the set of all finite diagonal matrices of R and by [8] every diagonal matrix D is equivalent to its Smith normal form (shortly SNF), so in any class of the equivalent elements in ∆(R) we can choose some SNF that represents this class in ∆(R)/ ∼. In fact, one can consider F (R) as a free abelian group generated by the classes of equivalence of all SNF of the matrices over R. The elements of the set ∆(R)/ ∼ we will denote as SNF(g), where g ∈ ∆(R).
Definition 5. The quotient group K ′ 0 (R) of a free abelian group F (R) by the subgroup generated by all expressions of the form SNF(g) + SNF(g ′ ) − SNF(g ⊕ g ′ ) we will call a weak Grothendieck group of a morphic ring R. The elements of K ′ 0 (R) will be denoted as [g]. In other words, K ′ 0 (R) is an abelian group of all classes of isomorphic finite direct sums of principal ideals of a morphic ring R with the following property: in F (R) if and only if m = l and there is a permutation π ∈ Sm such that ∀j : g j ∼ = g ′ π(j) .

Lemma 1. Two elements
After placing the summands with negative signs to the another part of the equality and using the previous remark we obtain that The lemma is proved.
In any case the computations that involves the elements of the group K ′ 0 (R) rise to the expressions with the principal ideals. So the first thing we need to know is: how the preimages of the equal elements of K ′ 0 (R) can be described in the terms of R? Lemma 2. Let R be a morphic ring and A, B, xR ∈ ∆(R) and A, B are reduced to the SNF. Then On the Weak Grothendieck Group of a Morphic Ring and its Representations [6] if A 1 and B 1 are the SNF of A ⊕ xR and B ⊕ xR then A 1 = B 1 . So we are going to compute explicitly SNF A 1 and B 1 using the Fitting invariants.
For the simplification of the notations in the proof below we will write a+b and ab for aR+bR and aR∩bR respectively. The ordering "≤" corresponds to the natural inclusion of the sets.
After computing the Fitting invariants the normal forms A 1 and B 1 are From the equality A 1 = B 1 we obtain the system of the principal ideal equations: Multiplying (in fact intersecting!) by an the equation bn + b n−1 x = an + a n−1 x we obtain an + anx = anbn + anb n−1 x = = anbn + bnb n−1 x = anbn + bnx ≤ bn.
But an = an +anx and so an ≤ bn. Analogously, multiplying the same equation by bn we will have that bn ≤ an. So an = bn.

Again, from the equation
So, the given system of the principal ideal equations simplifies and we have Again, we multiply the equation b n−1 + b n−2 x = a n−1 + a n−2 x by a n−1 and hence obtain a n−1 b n−1 + a n−1 b n−2 x = a n−1 If we multiply by x the equation bn + b n−1 x = an + a n−1 x we will obtain bnx + b n−1 x = anx + a n−1 x and hence b n−1 x = a n−1 x. Thus the equation a n−1 = a n−1 b n−1 + a n−1 b n−2 x implies that a n−1 ≤ b n−1 . Similarly b n−1 ≤ a n−1 . Therefore a n−1 = b n−1 . After the finite number of steps using the prescribed procedure we will have the following finally reduced system: But, multiplying the first equation by a 1 we obtain Again, by the similar consideration we can conclude that So, having SNFs of A and B such that A ⊕ xR ∼ = B ⊕ xR we have obtained that the summand xR can be cancellated and A = B as was desired. The lemma is proved.
As a corollary we obtain the following result.

respectively. Then by the previous lemma
Continuing this process we will finally obtain that A ∼ = B. If A and B are are reduced to the SNF then A = B. The theorem is proved. Now we need to formulate one well-known property of the tensor product of modules over the commutative ring. Proposition 1. Let M be a R-module and I, J are some ideals of a commutative ring R. Then In the following lemma we apply this Proposition in order to obtain one surprising property of the principal ideals of a morphic ring. Proof. By the definition of a morphic ring and the mentioned above result we have that aR In the classical K-theoretical investigations the Grothendieck's group K 0 (R) can be considered as a ring if we assume that R is a commutative ring and the product is defined as for any finitely generated projective R-modules P and Q over a commutative ring R. In the similar manner we obtain Theorem 2. Let R be a commutative morphic ring. Then an additive abelian group K ′ 0 (R) becomes a commutative ring with 1 if we define a product for any a, b ∈ R, and extend it on the arbitrary elements of K ′ 0 (R) by the linearity. Remark that any element any element of K ′ 0 (R) can be written as Now we try to understand how behaves K ′ 0 (R) under the base ring R change and how one can describe its structure in the simplest case.
0 is a functor from the category MorphicRings of the morphic rings to the Rings category.
Proof. Let f : R → R ′ be a homomorphism of the morphic rings. For any element Thus, a ring's map f : R → R ′ rises a correspondence f * : of the abelian groups K ′ 0 (R) and K ′ 0 (R ′ ). Moreover, if [aR], [bR] ∈ K ′ 0 (R) and additionally we set gR = aR ∩ bR, gR ∼ γR, aR ∼ αR, bR ∼ βR then So f * becomes a ring's homomorphism. Therefore .
Then we need to verify: is it a functor or not? Indeed, if f = 1 R : R → R then for any [ and hence K ′ − → R ′′ are two homomorphisms of the morphic rings then we need to prove that Without loss of the generality take any [aR] ∈ K ′ 0 (R) and assume that aR ∼ αR. Then as was desired. So, K ′ 0 is a functor. The proposition is proved.
As a consequence it can be shown that K ′ 0 preserves direct products of the morphic rings: Theorem 3. Let R be a morphic ring. Then K ′ 0 (R) has a direct summand isomorphic to the ring of integers Z.
Proof. Considering any maximal ideal M of R we define a natural homomorphism A map i * : is a monomorphism such that f * i * = 1 Z and the following short exact sequence The latter isomorphism proves the theorem.  a 2 , ..., a k ∈ M, a k+1  implies that [R] = [0]. By the Theorem 1 we obtain that R = 0 that is a contradiction, and such maximal ideal M cannot exist. After repeating the similar procedure to the other maximal ideals we obtain that there are no maximal ideals in R and R have to be a field. The theorem is proved.

The representation of K ′ 0 by the Witt vectors
In the current section we will try to find a convenient way for the addition and multiplication of the elements of K ′ 0 . Definition 5. A Witt ring (or Witt vectors) for a commutative ring R is called a set that is an abelian group under the multiplication operation between the formal power series (this operation represents the additive operation of a ring W (R)) and the ring multiplication operation is defined by the convolution rule in the following way: any f (t) ∈ W (R) can be written as so for the arbitrary r ∈ R we define and extend this rule to the infinite products. An identity of a ring W (R) is an element 1 + t and 1 is a zero element.
Before applying the mentioned ring construction we need to formulate the following definition. Other examples of globalizations can be obtained similarly. Thus, for any commutative Bezout ring R we can consider the ring completion G(R) = Ω −1 Ω(R) of its globalization Ω(R). Since G(R) becomes a commutative ring with identity then we can define a subsemiring of a ring W (G(R)) considered as a semiring.
Any element f (t) = 1 + (a 1 R)t + (a 2 R)t 2 + ... + (anR)t n ∈ W 0 (G(R)) can be expressed in the form An identity element of W 0 (G(R)) is 1 + Rt and 1 is its zero element. Furthermore, if are any elements of W 0 (G(R)) then their sum and product can be computed by the formulae After the direct computations one can conclude that the above definitions of the sum and product also belongs to W 0 (G(R)).
Theorem 5. If R is a morphic ring and G(R) = Ω −1 Ω(R) then where W ′ (R) is a ring completion of a semiring W 0 (G(R)).
Proof. In the following consideration the subtraction operation in the ring completion W ′ (G(R)) will be denoted by g(t). In fact, it is a formal polynomial's division. So, the ring W ′ (R) can be described as and multiplication for any a(t), b(t), c(t), d(t) ∈ W 0 (G(R)). We define a map .. ⊕ bmR are reduced to the SNF. The map F R is a bijection since the SNF is defined uniquely and a(t) and b(t) have a uniquely determined decompositions Also, it is a homomorphism since and Thus, F R is an isomorphism. The theorem is proved. Proof. If we set { for any homomorphism f : R → R ′ in the appropriate source category then images of f such as W ′ (f ), Ω(f ), G(f ) are precisely the homomorphisms in the target categories of the given maps. The fact that W ′ , Ω and G preserves identity homomorphisms and the compositions can be shown by the routine calculations. So, W ′ , Ω and G are the functors.

Theorem 6.
If we consider Ω as a functor Ω :

MorphicRings
Semirings then there is a natural equivalence of functors Proof. By Theorem 5 K ′ 0 (R) ∼ = W ′ (G(R)) for any morphic ring R via the isomorphism F R . So, if f : R → R ′ is any homomorphism of morphic rings R and R ′ then is a commutative diagram since Thus, K ′ 0 ≈ W ′ G as was desired. The theorem is proved. The latter result shows the way that one can compute the SNF of the block sum A⊕B and Kroneker's product A⊗B of two given matrices A and B that are already reduced to their SNF's. Since the multiplication in K ′ 0 (R) can be done after some number of the addition operations (this follows from the distributivity of the tensor product over the direct sums) then naturally arises a question: are there any other way to represent the elements of K ′ 0 (R) for more efficient evaluation of the sums of the given elements?
The answer is affirmative and below we give a solution.
that are reduced to their SNF then we represent [A] and [B] in a form · · · · · · · · · · · · · · · · · · · · · · · · 0 0 · · · 0 0 0 0 · · · xn x n−1 and multiply the respective matrices for [A] and [B] in M n+m (Ω(R)), then the resulting matrix will represent the sum [A ⊕ B]. In other words: if J is a Jordan matrix in M n+m (Ω(R)) with the zero eigenvalue then and the latter product will be the necessary result. On the other hand, it is not necessary to multiply pairwise every row and column in order to obtain the result. In fact, the situation can be solved even simpler -the product where T is a matrix whose rows are shifts of a row (an, a n−1 , · · · , a 1 , 1, 0, · · · 0). ∈ M n+m (Ω(R))

22
On the Weak Grothendieck Group of a Morphic Ring and its Representations

Suppose that
Since the highest terms of the LHS and RHS have to be equal then a 1 R∩a 2 R+b 1 R = a 1 ∩b 1 and hence Again, using the equality of the highest terms we have that The equality of the highest terms implies that a 2 R = b 1 R = b 2 R and hence the combination [ and hence . Since the both parts of the latter equality are already written in the SNF then b 1 R = aR ∩ b 1 R. Therefore b 1 R ⊆ aR and hence The number of terms in the LHS is m 2 + m and in the RHS is 2m. Since we have assumed that [B] is in the reduced form then the only possible case is m 2 +m = 2m and hence m = 1.
So, any idempotent After the substitution we will obtain Hence a 1 R = b 1 R and they can be cancellated in the expression [ In a case when R is a commutative von Neumann regular ring the structure of a ring K ′ 0 (R) becomes rather simple. Suppose that a ∈ R. Then there is x ∈ R such that a 2 x = a and as can easily see

Thus, if
for some [A ′ ] ∈ K ′ 0 (R). In other words, we have obtained the following result. The latter result is rather important since it has a connection with the usual Grothendieck's group K 0 (R) of a von Neumann regular ring R.
Theorem 7. If R is a commutative von Neumann regular ring then Proof. By [2,7,5,8] we know that 1) every finitely generated projective R-module is a finitely presented; 2) every finitely presented flat R-module is a projective; 3) every R-module over a von Neumann regular ring is flat 4) every commutative von Neumann regular ring is an elementary divisor ring 5) a commutative Bezout ring R is an elementary divisor ring if and only if any finitely generated R-module can be decomposed in a finite direct sum of some cyclic R-modules.
So, any A = a 1 R ⊕ ... ⊕ anR is a projective module since it is flat and any finitely generated projective module P can be decomposed in a finite direct sum of some cyclic modules. According to the construction of K 0 (R) and K ′ 0 (R) they have to coincide. The theorem is proved.
It is known that for any projective module P over a ring R there are some free R-module F and a submodule Q of F such that Thus as any element of K ′ 0 (R) can be expressed in form [P 1 ]− [P 2 ] then where P 2 ⊕ Q 2 ∼ = R N and we conclude that the studying of the structure of K ′ 0 (R) help to understand the Grothendieck's group K 0 (R).