Does the ring of regular expressions exist?

Question

It is well known that the set of Regular Expressions R over some alphabet form a semiring with:

• Concatenation as multiplication
• The empty string as the multiplicative identity
• 'Or' as addition
• The empty set of strings as the additive identity

Q. Does there exist an extension of R to a ring?

Answer 1

Regular languages on an alphabet $A$ form a semiring with union as addition (and the empty set as $0$) and concatenation product as product (and the language reduced to the empty word as $1$). This semiring is noncommutative if and only if $|A| > 1$.

This semiring can be identified with the semiring of rational series over the Boolean semiring $\mathbb{B} = \{0, 1\}$. As a set, this semiring can be embedded into the ring of rational series over the ring $\mathbb{Z}$, but this embedding is not a semiring embedding since in $\mathbb{B}$, $1 + 1 = 1$. A very good reference on this topic is [1].

[1] J. Berstel and C. Reutenauer, Noncommutative rational series with applications, Encyclopedia of Mathematics and its Applications, 137. Cambridge University Press, Cambridge, 2011. xiv+248 pp. ISBN 978-0-521-19022-0

Answer 2

The zero element is the empty set (i.e., empty language) and the unit element contains as its sole element the empty string. A ring requires to have an additive inverse for each element. But this is not the case when the operation is union of languages (i.e., union as sets).