## "as an algebra" or "as algebra" (zero article or indefinite article)?

3

This is a grammar question in the context of mathematics.

Introduction of the context: I use the word "algebra" here in the sense of a mathematical object (see Algebra over a field (Wikipedia)). Often an algebra is defined over a field K. One then speaks more precisely of a K-algebra or of an algebra over K.

So let K be a field. A K-algebra consists roughly of two things (the details do not matter here):

• a ring R
• a structure that relates it to K

In particular a ring R can become a K-algebra in different ways. So one needs to specify how a given ring R becomes a K-algebra. In such a situation I would write

"We consider R as an algebra over K via ...",

where in "..." I explain the structure that relates R to K.

Question: Should one use the indefinite article "an" in the previous phrase or not? Why?

2

Yes, you use the indefinite article an in your phrase an algebra. Without the article, 'algebra' would be read as an adjective rather than a noun.

In the structured phrase we consider X to be Y, the term Y is read as a property of X. E.g.: we consider an apple to be edible.

In your example, you're saying that X is one of potentially several kinds of Y, so an indefinite article is used. E.g.: we consider an apple to be a fruit.

Thanks for your reply. Reading "consider an apple to be a fruit", I'm not sure if we speak about the same thing. Let me try to explain a similar example: Consider the shape of the letters "M" and "W". Suppose you use a font where the forms of these letters would be identical, the only difference being that one is flipped. A child might encounter this form in elementary school. A teacher might say: "We can consider this form as a letter, either as (an) M or as (a) W." Both make this form a letter. Here "form" plays the role of "ring" and "letter" that of "algebra". Is this what you mean? – skew41 – 2015-11-03T14:53:08.500

@skew41 Yes, I think your form-M/W example is of the same structure as your ring-algebra example, and we are therefore speaking about the same thing. – Lawrence – 2015-11-03T15:01:02.707

@skew41 You could say this ring R is commutative since commutativity is a term that could be used to describe a ring. Or more to the point, you could say R is monoidal or R is a monoid. The difference is that monoidal is an adjective while monoid is a noun. Using another example, "this is fast" and "this is a car" are both valid phrases, but "this is car" is awkward. – Lawrence – 2015-11-03T15:10:09.580

I know the difference between an adjective and a noun. However, there are cases when nouns are used without article (so called "zero article"). My question was whether this is the case here. I often read "as algebra" as well as "as an algebra" in the context I described. Some mathematicians (I am not sure, but I guess native English speakers) even use both forms, sometimes "as algebra", sometimes "as an algebra". Therefore I am doubting. – skew41 – 2015-11-03T15:14:24.630

Examples from "Basic Algebra II" by Nathan Jacobson, second edition, 1989: "Hence, by the proof of Proposition 7.27, these elements generate GQ(R) as algebra over R/Q." and "We consider an extension field E/F of a field and regard this as an algebra over F." Maybe "as algebra" in the first phrase is used somehow as an adverb, since the verb "generate" needs explanation. – skew41 – 2015-11-03T15:23:08.423

Another quote some this book: "Since the radical of A as algebra or as ring is the intersection of the maximal left ideals, it is clear that rad A is the same set with the same addition and multiplication whether A is regarded as ring or as algebra. Similarly, A is primitive (semi- primitive) as algebra if and only if it is primitive (semi-primitive) as ring." – skew41 – 2015-11-03T15:28:34.453

1

@skew41 I'm not pointing out the difference between an adjective and a noun. I'm pointing out the difference between how adjectives and nouns are used in this context in English. From this link https://en.wikipedia.org/wiki/Nathan_Jacobson, it appears that Jacobson was not a native English speaker. While I don't know how fluent he was in English, the phrases "as algebra" and "as ring" in this context are consistent with the lack of articles in the Russian language (see http://esl.fis.edu/grammar/langdiff/russian.htm, near the bottom of the page).

– Lawrence – 2015-11-03T17:07:18.623

In fact, there are no articles in Russian, neither in Polish. Maybe I should have tried to give references from a native English speaker. Right now I do not find comparable ones by native English speakers. I hope that they serve at least as a motivation for my question. I am now inclined to believe that you are right and that this situation does not allow the "zero article". Thank you. – skew41 – 2015-11-03T22:06:20.293