Comment by aleph_minus_one

Rings:

* Determinant calculation:

- The Samuelson–Berkowitz algorithm is best understood in terms of general rings

- The Faddeev–LeVerrier algorithm and determinant calculation using Gaussian elimination work on rings with specific properties (for the Faddeev–LeVerrier algorithm the restriction is on the characteristic of the ring, for Gaussian elimination the ring must be an integral domain (ideally a field)).

* Ring-learning with errors (for post-quantum cryptography and homomorphic cryptography). Here, a specific ring is the central object.

* Number-Theoretic Transform (NTT): Basically a generalization of the Fourier Transform to the ring Z_n. Important for arbitrary-precision integer arithmetic

* Chinese Remainder Theorem. Often only formulated for the ring Z, but it can be generalized to larger classes of rings. Used for example in Shamir’s scheme for secret sharing (cryptography)

* The theory of BCH and Reed-Solomon codes uses a specific ring

* The AKS Primality Test (a really deep result in computational number theory) uses the ring Z_n[X]/(x^r-1).

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Algebras:

Very often, a ring is constructed from another ring. Examples:

* the polynomial ring R[X_1, ..., X_n]

* The ring of (square) matrices over a ring R

So, using algebras in algorithms often means: "we want to make use use of this additional structure that our (more sophisticated) ring has)". (Associative) R-algebras formalize this concept of "ring with additional structure".

To just give one algorithm for polynomials:

* Buchberger algorithm for computing a Gröbner basis

Other examples:

* Clifford algebras for a lot of geometric problems (special case: quaternions (a 4-dimensional \mathbb{R}-algebra) for rotations in \mathbb{R}^3).

* If you are willing to also consider semi-rings (in this case: tropical semi-rings): the Floyd-Warshall algorithm for finding shortest paths and the Viterbi algorithm for finding the most likely sequence of states in a Hidden-Markov Model (HMM) can very elegantly formulated using the matrix semiring over the tropical semiring.