а. Let I = {a + bi: a, b E Z[i]: 3 divides both a and b} . Prove that I is a maximal ideal of the ring Z[i] of Gaussian integers.
а. Let I = {a + bi: a, b E Z[i]: 3 divides both a and b} . Prove that I is a maximal ideal of the ring Z[i] of Gaussian integers.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![а.
Let I =
{a + bi: a, b E Z[i]: 3 divides both a and b} . Prove that I is a maximal ideal of
the ring Z[i] of Gaussian integers.
b. Let R be a ring with unity and ICR× R. Prove that I is an ideal of the ring R x R if and
only if I = 11 x I2 for some ideals I and I2 of R.
Prove that neither 2 nor 17 are prime elements in Z[i] (the ring of Gaussian integers).
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Transcribed Image Text:а.
Let I =
{a + bi: a, b E Z[i]: 3 divides both a and b} . Prove that I is a maximal ideal of
the ring Z[i] of Gaussian integers.
b. Let R be a ring with unity and ICR× R. Prove that I is an ideal of the ring R x R if and
only if I = 11 x I2 for some ideals I and I2 of R.
Prove that neither 2 nor 17 are prime elements in Z[i] (the ring of Gaussian integers).
С.
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