"\\mathbb{Z}[\\sqrt{2}]" is a ring under the ordinary addition and multiplication because:
1) "\\mathbb{Z}[\\sqrt{2}]" is closed under addition
If "x,y\\in \\mathbb{Z}[\\sqrt{2}]" , then "x=a+b\\sqrt{2}" and "y=c+d\\sqrt{2}" , where "a,b,c,d\\in \\mathbb{Z}" .
"x+y=(a+b\\sqrt{2})+(c+d\\sqrt{2})=(a+c)+(b+d)\\sqrt{2}\\in \\mathbb{Z}[\\sqrt{2}]"
2) addition is associative ( "\\mathbb{Z}[\\sqrt{2}]" is a subset of "\\mathbb{R}" )
3) there exists "0=0+0\\cdot \\sqrt{2}\\in \\mathbb{Z}[\\sqrt{2}]"
4) "\\forall x\\in \\mathbb{Z}[\\sqrt{2}],\\ x=a+b\\sqrt{2}, \\ \\ \\exists (-x)\\in \\mathbb{Z}[\\sqrt{2}], \\ -x=(-a)+(-b)\\sqrt{2}"
5) addition is commutative ("\\mathbb{Z}[\\sqrt{2}]" is a subset of "\\mathbb{R}" )
6) "\\mathbb{Z}[\\sqrt{2}]" is closed under multiplication
If "x,y\\in \\mathbb{Z}[\\sqrt{2}]" , then "x=a+b\\sqrt{2}" and "y=c+d\\sqrt{2}" , where "a,b,c,d\\in \\mathbb{Z}" .
"x\\cdot y=(a+b\\sqrt{2})(c+d\\sqrt{2})=(ac+2bd)+(ad+bc)\\sqrt{2}\\in \\mathbb{Z}[\\sqrt{2}]"
7) multiplication is associative ("\\mathbb{Z}[\\sqrt{2}]" is a subset of "\\mathbb{R}" )
8) distributivity ("\\mathbb{Z}[\\sqrt{2}]" is a subset of "\\mathbb{R}" )
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