The answer that I am going to give is implicitly contained in a few answers already given, but it is a bit too implicit, to my taste, so let me give it out and loud: Gröbner bases. When you solve a system of linear equations, you use Gaussian elimination, when you solve a system of polynomial equations of higher degrees, you use Gröbner bases, and it is very clear that solving systems of polynomial equations is something that people have to do for all sorts of applications.

That "very clear" is not just a belief held by a pure mathematicial: on a few occasions that I talked about something mathematical to people doing research in some real world questions of statistics, biology, engineering, Gröbner bases would be the only aspect of somewhat advanced algebra, not just commutative algebra, that they would have ever heard of. You can see some relevant bits of software solving applied problems in various areas here: http://www.risc-software.at/en/.

I can't resist from also saying that in some areas of pure maths, for a long time, saying the words "Gröbner bases" was a bit of faux pas, something that a true pure mathematician should rather leave as a discussion topic to people concerned with applications, something as silly and naive and so not worth mentioning as using a calculator to multiply two numbers. However, besides being a useful tool for computations, Gröbner bases and their generalisations also give methods to construct resolutions (starting from work of Anick in 1980s), and in particular to prove that a certain algebra (or an operad) is Koszul etc. So it certainly is something worth being aware of, really.

The answer that I am going to give is implicitly contained in a few answers already given, but it is a bit too implicit, to my taste, so let me give it out and loud: Gröbner bases. When you solve a system of linear equations, you use Gaussian elimination, when you solve a system of polynomial equations of higher degrees, you use Gröbner bases, and it is very clear that solving systems of polynomial equations is something that people have to do for all sorts of applications.

That "very clear" is not just a belief held by a pure mathematician: on a few occasions that I talked about something mathematical to people doing research in some real world questions of statistics, biology, engineering, Gröbner bases would be the only aspect of somewhat advanced algebra, not just commutative algebra, that they would have ever heard of. You can see some relevant bits of software solving applied problems in various areas here: http://www.risc-software.at/en/.

I can't resist from also saying that in some areas of pure maths, for a long time, saying the words "Gröbner bases" was a bit of faux pas, something that a true pure mathematician should rather leave as a discussion topic to people concerned with applications, something as silly and naive and so not worth mentioning as using a calculator to multiply two numbers. However, besides being a useful tool for computations, Gröbner bases and their generalisations also give methods to construct resolutions (starting from work of Anick in 1980s), and in particular to prove that a certain algebra (or an operad) is Koszul etc. So it certainly is something worth being aware of, really.