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Let $R$ be a commutative ring with unity, let $a$ be an element of $R$, and let $$\langle a \rangle := \{ ra \mid r \in R \}. $$ Then $\langle a \rangle$ is an ideal in $R$, called the principal ideal generated by the element $a$.

Now what is the most efficient way of pronouncing this symbol when reading a text on ring theory?

In particular, how best to read out loud expressions such as $F[x] / \langle p(x) \rangle$, where $F[x]$ is the integral domain of polynomials in an indeterminate $x$ with coefficients in a field $F$, $p(x)$ is a polynomial in $F[x]$, and of course $\langle p(x) \rangle$ is the principal ideal generated by $p(x)$?

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For the ideal generated by $p\in R[x]$, I would say "the ideal generated by $p$", or "the multiples of $p$". For $F[x]/(p)$ one could say "the ring of polynomials modulo$~p$" just as $\Bbb Z/n\Bbb Z$ can be pronounced "the integers modulo$~n$".

I would certainly drop the $(x)$ after the $p$, both from the notation and from the verbalised form. And maybe call the polynomial by a different letter to avoid possible confusion with a prime (irreducible element) of$~R$.

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Well, one solution would be to use the notation $I(a)$ for the ideal generated by $a$. Now if you keep the notation $\langle a \rangle$ you may say "Let brackets a be the ideal generated by $a$". For the loud expression, I would proceed in several steps:

  1. Let $F$ be a field.
  2. Let $F[X]$ be the ring of polynomials [in one indeterminate] with coefficients in $F$.
  3. Let $p \in F[X]$ be a polynomial.
  4. Let $F[X]/\langle p \rangle$ be the quotient of $F[X]$ by the ideal generated by $p$.
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