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?


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
    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
  • 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 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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