# Are there described algebras with "divisors of infinity"?

I wonder whether are there known algebraic systems where there are non-infinity (not satisfying $$x+a=x$$) elements such that their power or product is infinity (an algebraic object that satisfies $$x+a=x$$, like in extended reals)?

I can think about $$\overline{\mathbb R}^2$$, which seemingly has divisors of infinity like $$(1,\infty)$$, but what about those with elements that squared give infinity?

• Your question isn't very clear: (1) don't the extended reals already provide an example of what you are looking for? (2) what does $\overline{\Bbb{R}}^2$ mean? Jun 22 at 22:08
• @RobArthan oh, (1) is a typo, fixed. Jun 22 at 22:13
• @RobArthan (2)-extended reals, direct product with itself. Jun 22 at 22:14
• I am still not sure exactly what you are looking for (because I don't know what axioms you are expecting your algebras to satisfy). But how about $R = \{0, 1, 2, 3\}$ with addition defined by $\min(x + y, 3)$ and multiplication defined by $\min(xy, 3)$? Then the additive annihilator $\infty = 3$ and $2 \neq \infty$, but $2^2 = \infty$. Jun 22 at 22:26
• @Anixx -- to be clear, you're looking for a system with $+$ and $\times$, where $a + \infty = \infty$ for all $a$, and where there's an element which squares to $\infty$? Jun 22 at 22:40

Such structure $$(S,+,\cdot)$$ exists.

Take any non trivial ring $$A$$ with unit having an element $$e\in A$$ such that $$e^2=0_A$$.

Set $$S=A$$, $$+=\cdot_A$$, $$\cdot=\cdot_A$$, $$\infty=0_A$$.

Then for all $$a\in A$$, $$a+\infty=a\cdot 0_A=0_A=\infty$$, and $$e\cdot e=e^2=0_A=\infty$$. However, not all element is $$\infty$$. Indeed, $$1_A\neq \infty$$. Otherwise, $$1_A=0_A$$ and $$A$$ would be trivial.

As a concrete example, you may take $$A=\mathbb{Z}/n^2\mathbb{Z}$$, and $$e=\bar{n}$$ (there are other examples as well in the same spirit)

However, I don't understand why you are looking for such structures. They have extremely limited interest from a computational point of view. For example $$(S,+)$$ cannot be a group (because of the existence of $$\infty$$), and I am pretty sure that you cannot have distributivity properties, even if a proof does not come into my mind right away...

• Thanks for the answer. I still cannot understand some notation though. What is $\overline{n}$ for instance? And yes, I also thought that distributivity is not possible there. I have two structures of this kind in my mind (for the both I already have code in Mathematica defining multiplication). They are the first two examples via this link: math.stackexchange.com/questions/4462705/… That's the motivation. 2 days ago
• Another motivation is to have simple structure that would describe divergent formal power series and integrals, I thought that divisors of infinity may play a role of these objects, just like dual unity plays a role of infinitesimal. 2 days ago
• well, if $m$ is an integer, $\bar{m}$ is it class modulo $n^2$. 2 days ago