Precisely, if an R-module M has a finite presentation, and R^k → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated?

Basically I want to believe I can choose generators for M however I please, and still get a finite presentation. I have reasons from algebraic geometry to believe this, but it seems like a very basic result, so I would like to understand it directly in terms of the commutative algebra, which I just can't seem to figure out...

(Here R is an arbitrary commutative ring, with no other hypotheses.)

Edit: All maps here are maps of R-modules. Also, the reason this is not the same as "does finite presentation imply coherent?" is that I am only asking for finite type kernels of surjections R^k → M. That the hypotheses assume surjectivity is a common misreading of the general definition of "coherent".

If the answer to the above is "yes", then coherent will mean "finite type, and all finite type submodules are finite presentation"

Precisely, if an R-module M has a finite presentation, and R^k → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated?

Basically I want to believe I can choose generators for M however I please, and still get a finite presentation. I have reasons from algebraic geometry to believe this, but it seems like a very basic result, so I would like to understand it directly in terms of the commutative algebra, which I just can't seem to figure out...

(Here R is an arbitrary commutative ring, with no other hypotheses.)

Edit: All maps here are maps of R-modules. Also, the reason this is not the same as "does finite presentation imply coherent?" is that I am only asking for finite type kernels of surjections R^k → M. That the hypotheses assume surjectivity is a common misreading of the general definition of "coherent".

If the answer to the above is "yes", then coherent will mean "finite type, and all finite type submodules are finite presentation"