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The degree means the dimension of the ambient module. For example, the vector space spanned by (2,0,2) inside R^3 is one dimensional but is of degree 3.
In your case, you have a module over the integers, perhaps spanned by (2) inside of a rank one free Z-module. When you reduce mod 2, the spanning vector becomes (0), so the resulting module over GF(2) is zero dimensional.
