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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?

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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.

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  • 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
    Nov 3 '15 at 14:53
  • @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
    Nov 3 '15 at 15:01
  • @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
    Nov 3 '15 at 15:10
  • 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
    Nov 3 '15 at 15:14
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    @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 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 esl.fis.edu/grammar/langdiff/russian.htm, near the bottom of the page).
    – Lawrence
    Nov 3 '15 at 17:07

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