‡ Every function on a finite residue class ring D/I of a Dedekind domain D is induced by an integer-valued polynomial on D that preserves congruences mod I if and only if I is a power of a prime ideal. If R is a finite commutative local ring with maximal ideal P of nilpotency N satisfying for all a, b∈R, if ab∈ P then a∈ P k , b∈ P j with k+ j ≥min(n,N), we determine the number of functions (as...

so that every square is covered by exactly one domino? In other words, can R be tiled by vertical and horizontal dominoes? The coloring gives the answer to this well-known problem away. The region R has 32 black squares and 30 white squares. Since each domino covers exactly one black and one white square, no tiling is possible. The aim of this article is to explain a way to tackle tiling proble...

A ring is a set R with two binary operations usually called addition, denoted as +, and multiplication, denoted as ·, such that (R, +) satisfies the five axioms of closure, associativity, commutativity, identity element (called zero, 0), and inverse element; and (R, ·) satisfies the three axioms of closure, associativity, and identity element (called one, 1). Furthermore, multiplication (·) dis...

Definition 1.1 (Rings). The algebraic structure “ring” R is a set with two binary operations + and ·, respectively named addition and multiplication, satisfying • (R,+) is an abelian group (i.e. a group with commutative addition), • is associative (i.e. 8a, b, c 2 R, (a · b) · c = a · (b · c)) , • and the distributive law holds (i.e. 8a, b, c 2 R, (a+ b) · c = a · c+ b · c, a · (b+ c) = a · b+ ...

Let G be a group and R G-graded ring. In this paper, we present examine the concept of graded weakly 2-absorbing ideals as in generality prime ring which is not commutative, demonstrates that symmetry obtained lot outcomes commutative rings remain are commutative.