Return to Answer

This was something that I found very, very confusing when I first began studying factorization in rings with zero-divisors. Nearly every author uses the term "irreducible" and "associate"; however, the problem is they mean different things in each case. As far as I can tell D.D. Anderson and A. Frazier's method of 3 choices for associate, and several choices of irreducible (irreducible, strongly irreducible, m-irreducible, very strongly irreducible) is the most general and encompasses the others in the following way:

In Galovich's paper he chooses for associate the relation $\approx$. That is $a$ and $b$ are "associate" in the Galovich sense if $a=\lambda b$ for some unit $\lambda$. His choice for "irreducible", is what Anderson and Frazier termed "very strongly irreducible."

In Bouvier's papers we have "associate" is $\sim$ as in Anderson and Frazier $a$ and $b$ are associate if $(a)=(b)$. However, the choice for "irreducible" is what is called m-irreducible, which comes from the fact that $a$ is m-irreducible if it is maximal among all principal ideals. For non-zero elements, the strength of this type of irreducible falls between strong irreducible and very strong irreducible. For zero, $0$ is m-irreducible if and only if $R$ is a field. For zero to be irreducible, strongly irreducible, or very strongly irreducible is equivalent to $R$ being a domain.

This was something that I found very, very confusing when I first began studying factorization in rings with zero-divisors. Nearly every author uses the term "irreducible" and "associate"; however, the problem is they mean different things in each case. As far as I can tell D.D. Anderson and Valdez-Leon's method of 3 choices for associate, and several choices of irreducible (irreducible, strongly irreducible, m-irreducible, very strongly irreducible) is the most general and encompasses the others in the following way:

In Galovich's paper he chooses for associate the relation $\approx$. That is $a$ and $b$ are "associate" in the Galovich sense if $a=\lambda b$ for some unit $\lambda$. His choice for "irreducible", is what Anderson and Valdez-Leon termed "very strongly irreducible."

In Bouvier's papers we have "associate" is $\sim$ as in Anderson and Valdez-Leon $a$ and $b$ are associate if $(a)=(b)$. However, the choice for "irreducible" is what is called m-irreducible, which comes from the fact that $a$ is m-irreducible if it is maximal among all principal ideals. For non-zero elements, the strength of this type of irreducible falls between strong irreducible and very strong irreducible. For zero, $0$ is m-irreducible if and only if $R$ is a field. For zero to be irreducible, strongly irreducible, or very strongly irreducible is equivalent to $R$ being a domain.

EDIT: I just wanted to add a few mathematical comments to respond to your mention of Galovich's paper.

This was something that I found very, very confusing when I first began studying factorization in rings with zero-divisors. Nearly every author uses the term "irreducible" and "associate"; however, the problem is they mean different things in each case. As far as I can tell D.D. Anderson and A. Frazier's method of 3 choices for associate, and several choices of irreducible (irreducible, strongly irreducible, m-irreducible, very strongly irreducible) is the most general and encompasses the others in the following way:

In Galovich's paper he chooses for associate the relation $\approx$. That is $a$ and $b$ are "associate" in the Galovich sense if $a=\lambda b$ for some unit $\lambda$. His choice for "irreducible", is what Anderson and Frazier termed "very strongly irreducible."

In Bouvier's papers we have "associate" is $\sim$ as in Anderson and Frazier $a$ and $b$ are associate if $(a)=(b)$. However, the choice for "irreducible" is what is called m-irreducible, which comes from the fact that $a$ is m-irreducible if it is maximal among all principal ideals. For non-zero elements, the strength of this type of irreducible falls between strong irreducible and very strong irreducible. For zero, $0$ is m-irreducible if and only if $R$ is a field. For zero to be irreducible, strongly irreducible, or very strongly irreducible is equivalent to $R$ being a domain.

There is yet another type of atomic referred to as i-atomic which is in terms of factorization into products of principal ideals, and I believe this is equivalent to the U-factorizations studied by Axtell, and Fletcher.

As you can see this is perhaps why most people skip the factorization with zero-divisors present. Things get complicated rather quickly, but as the student alluded, perhaps this is what makes it more interesting in some ways.

This might be way more information than you want, but I just came across this site and it seems really exciting and useful and I saw your question was about what I study, and I may have gone a little overboard.

The biggest problem complicating factorization when there are zero-divisors present in my experience stems from the fact that there are several ways to define "associate" that are no longer equivalent.

For instance, the reference given by Guntram provides a good introduction to the issues:

$a$ and $b$ are associates, written $a\sim b$ iff $(a)=(b)$

$a$ and $b$ are strong associates, written $a \approx b$ iff $a=\lambda b$ for some $\lambda \in U(R)$

$a$ and $b$ are very strong associates, written $a\cong b$ iff $(a)=(b)$ and if $a=rb$ for some $r\in R$, then $r$ is a unit.

Each gives rise to a different type of irreducible along the lines of if $a=a_1 \cdots a_n$ is a factorization then $a \sim a_i$, $a \approx a_i$, or $a \cong a_i$ for some $1 \leq i \leq n$, respectively $a$ is irreducible, strongly irreducible, or very strongly irreducible.

Examples are given to show how the different types of irreducibles are no longer equivalent.

This means when looking at something like how to define a Unique Factorization Ring (instead of domain) you need to pick what time of atoms you want your factorizations to be broken into and then when rearranging up to associate, what to pick for that.

Of course, this is only part of the story. Unique Factorization Rings are pretty nice, there are weaker properties that will fail by the existence of zero-divisors, idempotent elements and nilpotent elements.

Zero-divisors mean, you can factor zero: Say $ab=0$. Well so does $0=ab=ababab=abxyzw$ or anything after it has already become zero.

Or nilpotents: $x^n=0$ implies $0=(x^{n-1})^{i}$ for all $i \geq 2$.

So do you allow zero to be factored? Maybe just don't allow zero to appear as a factor? Even that is not enough to deal with it completely.

Suppose $e=e^2$, well then $e=e^i$ for all $i\geq 1$, so does that mean your ring is not even a Bounded Factorization Ring? This is what happened in your example $\mathbb{Z}/6 \mathbb{Z}$, $4$ is idempotent.

One approach that seems particularly nice was introduced by Fletcher in two papers in Proc. Cambridge Philos. Soc. in 1969 and 1970 called Unique Factorization Rings and The structure of Unique Factorization Rings. This was U-factorization where you have essential and inessential divisors to help deal with things like nilpotents and idempotents. M. Axtell has a couple of nice papers on it too in 2002 and 2003 U-factorizations in Commutative Rings with zero-divisors and Properties of U-factorizations.

This makes something like $\mathbb{Z}/6 \mathbb{Z}$ a U-UFR, U-HFR, U-BFR, etc, but not a UFR, HFR, or a BFR.

Interesting stuff in my opinion.

The biggest problem complicating factorization when there are zero-divisors present in my experience stems from the fact that there are several ways to define "associate" that are no longer equivalent.

For instance, the reference given by Guntram provides a good introduction to the issues:

$a$ and $b$ are associates, written $a\sim b$ iff $(a)=(b)$

$a$ and $b$ are strong associates, written $a \approx b$ iff $a=\lambda b$ for some $\lambda \in U(R)$

$a$ and $b$ are very strong associates, written $a\cong b$ iff $(a)=(b)$ and if $a=rb$ for some $r\in R$, then $r$ is a unit.

Each gives rise to a different type of irreducible along the lines of if $a=a_1 \cdots a_n$ is a factorization then $a \sim a_i$, $a \approx a_i$, or $a \cong a_i$ for some $1 \leq i \leq n$, respectively $a$ is irreducible, strongly irreducible, or very strongly irreducible.

Examples are given to show how the different types of irreducibles are no longer equivalent.

This means when looking at something like how to define a Unique Factorization Ring (instead of domain) you need to pick what type of atoms you want your factorizations to be broken into and then when rearranging up to associate, what to pick for that.

Of course, this is only part of the story. Unique Factorization Rings are pretty nice, there are weaker properties that will fail by the existence of zero-divisors, idempotent elements and nilpotent elements.

Zero-divisors mean, you can factor zero: Say $ab=0$. Well so does $0=ab=ababab=abxyzw$ or anything after it has already become zero.

Or nilpotents: $x^n=0$ implies $0=(x^{n-1})^{i}$ for all $i \geq 2$.

So do you allow zero to be factored? Maybe just don't allow zero to appear as a factor? Even that is not enough to deal with it completely.

Suppose $e=e^2$, well then $e=e^i$ for all $i\geq 1$, so does that mean your ring is not even a Bounded Factorization Ring? This is what happened in your example $\mathbb{Z}/6 \mathbb{Z}$, $4$ is idempotent.

One approach that seems particularly nice was introduced by Fletcher in two papers in Proc. Cambridge Philos. Soc. in 1969 and 1970 called Unique Factorization Rings and The structure of Unique Factorization Rings. This was U-factorization where you have essential and inessential divisors to help deal with things like nilpotents and idempotents. M. Axtell has a couple of nice papers on it too in 2002 and 2003 U-factorizations in Commutative Rings with zero-divisors and Properties of U-factorizations.

This makes something like $\mathbb{Z}/6 \mathbb{Z}$ a U-UFR, U-HFR, U-BFR, etc, but not a UFR, HFR, or a BFR.

Interesting stuff in my opinion.