The tropical semi-ring is the triplet (R,⊕, ) with x⊕ y := min(x,y) and x y := x+ y . Its algebraic properties are somewhat modest, if one compares it with rings and fields, which are more common in most areas of mathematics. Still the distributive laws hold, and as an additional catch the addition is idempotent, that is, x⊕ x = x holds for all x ∈ R. As defined our tropical semi-ring is lackin...

Decomposing a given matrix into two factor matrices is a frequently used technique in data mining for uncovering underlying latent patterns in the data. Unlike in pure mathematics, the emphasis is put on obtaining results that are interpretable, rather than necessarily having a small reconstruction error. One approach to increase interpretability is to pose constraints on the factors. For examp...

Let R be a commutative ring with identity and M an R–module. If M is either locally cyclic projective or faithful multiplication then M is locally either zero or isomorphic to R. We investigate locally cyclic projective modules and the properties they have in common with faithful multiplication modules. Our main tool is the trace ideal. We see that the module structure of a locally cyclic proje...

The double-precision floating-point arithmetic, specifically multiplication, is a widely used arithmetic operation for many scientific and signal processing applications. In general, the double-precision floating-point multiplier requires a large 53 × 53 mantissa multiplication in order to get the final result. This mantissa multiplication exists as a limit on both area and performance bounds o...

In recent work we called a ring R a GGCD ring if the semigroup of finitely generated faithful multiplication ideals of R is closed under intersection. In this paper we introduce the concept of generalized GCD modules. An R-moduleM is a GGCD module if M is multiplication and the set of finitely generated faithful multiplication submodules of M is closed under intersection. We show that a ring R ...

All rings are commutative with identity and all modules are unital. The tensor product of projective (resp. flat, multiplication) modules is a projective (resp. flat, multiplication) module but not conversely. In this paper we give some conditions under which the converse is true. We also give necessary and sufficient conditions for the tensor product of faithful multiplication Dedekind (resp. ...

Algebra extensions A ⊆ B where A is a left B-module such that the Baction extends the multiplication in A are ubiquitous. We encounter examples of such extensions in the study of group actions, group gradings or more general Hopf actions as well as in the study of the bimodule structure of an algebra. In this paper we are extending R.Wisbauer’s method of constructing the central closure of a se...

Algebra extensions A ⊆ B where A is a left B-module such that the Baction extends the multiplication in A are ubiquitous. We encounter examples of such extensions in the study of group actions, group gradings or more general Hopf actions as well as in the study of the bimodule structure of an algebra. In this paper we are extending R.Wisbauer’s method of constructing the central closure of a se...

In this paper by considering the notion of BCK-module X, we introduce new concept which is named extended BCK-module or briefly X-module and we obtain some interesting results by it. Furthermore, we define the notion of multiplication X-module. 2000 Mathematics subject classification: 06D99 • 08A30