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Let $R$ be a non-unital ring. Let $F:R\times R\longrightarrow R$ be a function given by the formula $F(x,y)=xy.$ Let $r\not\in\operatorname{im}(F).$ Such elements can exists, for example $2\in 2\mathbb Z$ isn't a product. It seems to be a major difference between unital and non-unital rings. I'm only starting to study non-unital rings and I thought it would be a good idea to understand this phenomenon better first. But I don't know any terminology, whence my question. What is the name (if there is any) of an element such as $r?$ Is there always such an element in a ring that actually doesn't have a unity? If not, what is the name of a ring in which such an element exists?

And finally, where can I read about it?

2 Answers2

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Consider the rng of cofinitely zero infinituples over $C$ or $R$. I.e. the set of all infinituples that are entirely zero after a while. This doesnt have a multiplicative identity but every element can be expressed as a product of rng elements.

As for the terminology or References, I remain woefully ignorant.

Arkady
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  • Wait, how is this set a ring? What is the product of say, a length 10 sequence and a length 20 sequence? I could see concatenation as a product, but then what is their sum? Or do we put an infinite string of 0s at the end of the sequence? – 2'5 9'2 Jul 15 '12 at 00:17
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    @alex.jordan Just imagine everything as an infinituple. I.e. concatenate zeroes after every finite sequence. Then addition and multiplication works the obvious way. And it's a rng, not a ring. Its missing the identity, you see. – Arkady Jul 15 '12 at 00:19
  • I see - in time to edit my last comment :) By the way, I purposely did not use the abhorrent term "rng", which just looks like a typo. Someone at some point thought they were being clever - I just think they were being boneheaded. "Nonunital ring" works for me. – 2'5 9'2 Jul 15 '12 at 00:21
  • @alex.jordan Surely you admit that the term "rng" is rather clever. Perhaps I am too easily amused.:) – Arkady Jul 15 '12 at 00:26
  • Spoken out loud, it sounds too similar to "ring". Typed, I am left wondering if "ring" was intended, and there is a typo. So it is not good terminology. – 2'5 9'2 Jul 15 '12 at 00:28
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Suppose you have an algebra $A$ with an augmentation $\epsilon : A \to k$. Let $I$ be the kernel of $\epsilon$. Then elements that can not be written as products, can be thought of as elements of $I/I^2$ and are called indecomposable.

I am sure there are plenty of books, but I don't know about any specific books.

Sean Tilson
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  • A (wild) (educated) guess in relation to arithmetic tells me that they could be called irreducible, or even prime, or indecomposable, as Sean Tilson suggested. That is, if we make divisibility more abstract (i.e. require divisibility in a rng, but not thinking about gcd's or lcm's...) – Adrian Manea Jan 27 '12 at 17:33
  • I don't like the term "indecomposable." In a ring with unit, you'd want it to refer to an element which can't be written as the product of two non-unital elements. – Qiaochu Yuan Jan 27 '12 at 17:46
  • Perhaps the term simple element could be used. I think it conjures up the appropriate association without it having some other (element-wise) meaning. – Miha Habič Jan 27 '12 at 18:26
  • @QiaochuYuan: Isn't that what I have written? The elements of $I$ are non-unital if there is a unit and the composition of $\epsilon$ with the unit map is the identity of $k$. – Sean Tilson Jan 28 '12 at 03:53