LetBbe a commutative ring with identity,m,n, andrbe positive integers such thatr≤ min{m,n},a1, …,ar(resp.b1, …br) be integers such that 1 ≤a1< … <ar≤m(resp. 1 ≤b1< … <br<n) andU(resp.V) be the most generalm×r(resp.r×n) matrix such thats-minors of firstas− 1 rows (resp.bs− 1 columns) ofU(resp.V) are all zero fors= 1, …,r. We investigate theB-algebraCgenerated by all the entries ofUVand all ther-minors ofUandV. We introduce a Hodge algebra structure, to which the discrete Hodge algebra associate is Cohen Macaulay, onCand prove thatCis Cohen-Macaulay if so isB. Using this Hodge algebra structure, we show thatCis the ring of absolute invariants of a certain group action, compute the divisor class group and the canonical class ofC, and give a criterion of Gorenstein property ofCin terms ofa1,…,arandb1…,br.