Uses of algebraic structures in theoretical computer science
My impression is that, by and large, traditional algebra is rather too specific for use in Computer Science.  So Computer Scientists either use weaker (and, hence, more general) structures, or generalize the traditional structures so that they can fit them to their needs.  We also use category theory a lot, which mathematicians don't think of as being part of algebra, but we don't see why not.  We find the regimentation of traditional mathematics into "algebra" and "topology" as separate branches inconvenient, even pointless, because algebra is generally first-order whereas topology has a chance of dealing with higher-order aspects.  So, the structures used in Computer Science have algebra and topology mixed in.  In fact, I would say they tend more towards topology than algebra.  Regimentation of reasoning into "algebra" and "logic" is another pointless division from our point of view, because algebra deals with equational properties whereas logic deals with all other kinds of properties as well.

Coming back to your question, semigroups and monoids are used quite intensely in automata theory.  Eilenberg has written a 2-volume collection, the second of which is almost entirely algebra.  I am told that he was planning four volumes but his age did not allow the project to be finished.  Jean-Eric Pin has a modernized version of a lot of this content in an online book.  Automata are "monoid modules" (also called monoid actions or "acts"), which are at the right level of generality for Computer Science.  Traditional ring modules are probably too specific.

Lattice theory was a major force in the development of denotational semantics.  Topology was mixed into lattice theory when Computer Scientists, jointly with mathematicians, developed continuous lattices and then generalized them to domains.  I would say that domain theory is Computer Scientists' own mathematics, which traditional mathematics has no knowledge of.

Universal algebra is used for defining algebraic specifications of data types.  Having gotten there, Computer Scientists immediately found the need to deal with more general properties: conditional equations (also called equational Horn clauses) and first-order logic properties, still using the same ideas of universal algebra.  As you would note, algebra now merges into model theory.

Category theory is the foundation for type theory.  As Computer Scientists keep inventing new structures to deal with various computational phenomena, category theory is a very comforting framework in which to place all these ideas.  We also use structures that are enabled by category theory, which don't have existence in "traditional" mathematics, such as functor categories.  Also, algebra comes back into the picture from a categorical point of view in the use of monads and algebraic theories of effects.  Coalgebras, which are the duals of algebras, also find a lot of application.

So, there is a wide-ranging application of "algebra" in Computer Science, but it is not the kind of algebra found in traditional algebra textbooks.

Additional note: There is a concrete sense in which category theory is algebra.  Monoid is a fundamental structure in algebra.  It consists of a binary "multiplication" operator that is associative and has an identity.  Category theory generalizes this by associating "types" to the elements of the monoid, $a : X \rightarrow Y$.  You can "multiply" the elements only when the types match: if $a : X \rightarrow Y$ and $b : Y \to Z$ then $ab : X \to Z$.  For example, $n \times n$ matrices have a multiplication operation making them a monoid.  However, $m \times n$ matrices (where $m$ and $n$ could be different) form a category.  Monoids are thus special cases of categories that have a single type.  Rings are special cases of additive categories that have a single type.  Modules are special cases of functors where the source and target categories have a single type.  So on.  Category theory is typed algebra whose types make it infinitely more applicable than traditional algebra.