When are Associates Unit Multiples?

Algebraic properties of expectation semirings

“…If n = 0, then (t, n) = (0, 0) and we are done. Otherwise As with module theory [3], we give the following definition: It is, then, easy to verify that ((s, 0)) = ((t, 0)). Therefore, (s, 0) ∼ (t, 0).…”

An Algebraic Framework for End-to-End Physical-Layer Network Coding

“…Throughout the paper R denotes a finite PIR and (r) denotes the ideal generated by r ∈ R. Recall that elements a, b ∈ R generate the same ideal if and only if they are associate, i.e., there exists an invertible element ε ∈ R with a = εb (see e.g. [1]). An element g ∈ R is a greatest common divisor (gcd) of a 1 , ..., a s ∈ R if and only if (a 1 )+(a 2 )+.…”

“…Let R = Z 8 , whose ideals 0 (4) (2) (1) have canonical generators 0, 4, 2, 1. Choose 0, 1, 2, 3 as canonical representatives for residue classes modulo (4), 0, 1 as canonical representatives for residue classes modulo (2), and 0 as canonical representative for the residue class modulo (1). The rows of the following matrices generate the same R-module.…”

Generalized factorization in commutative rings with zero-divisors

“…As in [1], a ring R is said to be strongly associate (resp. very strongly associate) ring if for any a, b ∈ R, a ∼ b implies a ≈ b (resp.…”

“…There are two approaches to ensuring this can be said: (1) insist that our ring R is reduced so there are no non-trivial nilpotent elements or (2) define a modification of τ z to be τ ∆ z := τ z − ∆ ∩ (Nil(R) × Nil(R)), that is aτ ∆ z b ⇔ ab = 0 and a = b. Both of these choices result in having no repeated factors in any given τ ∆ z (τ z )-factorization (in a reduced ring) which will be useful in several of the proofs.…”