Is the term “adeal” ever used in ring theory

Question

I'm making my way through Burton's A First Course in Rings and Ideals (1970). In this book, a ring need not have an identity. Anyway, in problem 27 of the second chapter (p.38), he introduces the concept of an adeal:

A nonempty subset $A$ of a ring $R$ is termed an adeal of $R$ if
(i) $a, b \in A$ imply $a + b \in A$,
(ii) $r \in R$ and $a \in A$ imply both $ar \in A$ and $ra \in A$.

When, however, I perform the obvious searches on Google for this term, all I come up with is Burton's book.

Is this concept defunct?
Does it go by another name?

For the person who wanted it, here is the question:

Prove that
a) An adeal $A$ of $R$ is an ideal of $R$ if for each $a \in A$ there is an integer $n \neq 0$, depending upon $a$, such that $na \in aR + Ra$. (This condition is satisfied, in particular, if $R$ has a multiplicative identity.)
b) Whenever $R$ is a commutative ring, the condition in part (a) is a necessary as well as sufficient condition for an adeal to be an ideal.

screenshot of text in case of typos

Answer 1

Searching MathSciNet for "adeal" anywhere (including both titles and reviews) produced just one result:

MR1532157 Luh, Jiang; Classroom Notes: Ideals and Adeals of a Ring. Amer. Math. Monthly 70 (1963), no. 5, 548–550.

So the answer to the title of the question, is it ever used, is yes. But it's not used much; in particular, it's not used enough for me to have heard of it before this question.