Step 1
A ring R is a set with two binary operations, addition and multiplication, satisfying several properties: R is an Abelian group under addition, and the multiplication operation satisfies the associative law
"a(bc)=(ab)c"
and distributive laws
"a(b+c)=ab+ac"
and
"(b+c)a=ba+ca"
for every
"a,b,c \\in R"
The identity of the addition operation is denoted 0. If the multiplication operation has an identity, it is called a unity. If multiplication is commutative, we say that R is commutative.
A subring of a ring R is a subset
"S \\subseteq R"
that is a ring under the operations of R.
Let A be a subring of a ring R. If, for every "a \\in A" and "r \\in R"
"ar \\in A"
and
"ra \\in A,"
we say that A is an ideal of R.
The ring "Z[i]" of Gaussian integers is defined as
"{a+bi\/ a,b \\in Z}"
under ordinary addition and multiplication of complex numbers.
Step 2
The Subring Test states that a nonempty subset S of a ring R is a subring if and only if it is closed under subtraction and multiplication; that is, if
"a,b \\in S,"
then,
"a-b, ab \\in S"
Let
"a+bi, c+di \\in S"
Step 3
"(a+bi)-(c+di)= (a-c)+(b-d)i"
Since
"a+bi, c+di \\in S"
both b and d are even. Hence "b-d" is even, and
"(a+bi)-(c+di) \\in S"
Step 4
Similarly,
"(a+bi)( c+di) =ac+adi+bci-bd \\implies (ac-bd)(ad+bc)i"
Step 5
Since b and d are even, "ad" and "bc", are even, and so is "ad+bc". It follows that
"(a+bi)( c+di) \\in S"
Step 6
However,
"I \\in S"
and
"Ii=i \\notin S."
We conclude that S is not an ideal of "Z[i]."
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