The Macaulay2 engine implements rings, elements in these rings, matrices over these rings, mutable matrices, ring maps, and computations including

Groebner bases
Hilbert functions
free resolutions

The engine is written in C++ over a long period of time, some of it before the c++ standard library was available or robust. The engine is being updated to c++17, and eventually to c++20.

See also: Rings; Matrices; Computations