Tag Archives: factorisation

Image factorisation in abelian categories

Let $R$ be a ring and $f:B\to C$ be a morphism of $R$-modules. The image of $f$ is of course $${\rm im}(f) = \{ f(x) : x\in B \}.$$The image of $f$ is a submodule of $C$. It is pretty much self-evident that $f$ factors as $$B\xrightarrow{e} {\rm im}(f)\xrightarrow{m} C$$where $e$ is a surjective homomorphism […]

Non-unique Factorisation: Part 2

We are continuing the series on non-unique factorisation. For a handy table of contents, visit the Post Series directory. In Part 1 of this series, we introduced for a commutative ring three types of relations: Associaties: $a\sim b$ means that $(a) = (b)$ Strong associates: $a\approx b$ means that $a = ub$ for $u\in U(R)$ […]

Non-unique Factorisation: Part 1

If $F$ is a field then the polynomial ring $F[x]$ is a unique factorisation domain: that is, every nonunit can be written uniquely as a product of irreducible elements up to a unit multiple. So in $\Q[x]$ for example, you can be sure that the polynomial $x^2 – 2 = (x-2)(x+2)$ can't be factored any […]

Generators of Principal Ideals and Unit Association

Let us ponder a familiar property of any integral domain $R$. Suppose that $a,b\in R$ are such that $(a) = (b)$, where $(a)$ denotes the ideal generated by $a$. Then $a = rb$ and $b = sa$ for some elements $r,s\in R$. Combining these two facts together […]