Question: Consider the ring R = Z18. Find all the units in R. Find all the zero

Question

Question: Consider the ring R = Z18. Find all the units in R. Find all the zero divisors in R. Find all the maximal ideals in R. Find all the prime ideals in R. What is the characteristic of R? Is R an integral domain?

Answer:
Units of Z18: [1], [5], [7], [11], [13], [17] [How do I get units?]

All nonzero elements, that are not units, are zero divisors in Z18: [2], [3], [4], [6], [8], [9], [10], [12], [14], [15], [16]. [How do I get that these are the zero divisors] In particular, Z18 is not an integral domain. [How do I get that Z18 is not an integral domain]

char(Z18) = 18. [characteristic of R? Just to make sure]

In Z18 all prime ideals are maximal [How do I know all the prime ideals are maximal and how do I find], they are principal ideals: [2]R(= [4]R = [8]R = [10]R = [14]R = [16]R), [3]R(= [15]R) [How did I find these ideals]

Please explain.

Explanation / Answer

Consider the ring R = Z18.

The units in R are precisely those elements which are relatively prime to 18, ie., those elements, say 'a' for which gcd(a,18) = 1. Thses are 1, 5,7,11,13,17 as all these numbers are relatively prime to 18.

We know that a unit is not a zero divisor. Thus the contrapositive gives, if a zero divisor, then a non unit. Thus the elements other than units mentioned above, all comes as zero divisors in R = Z18.ie., 2,3,4,6,8,9,10,12,14,15,16.

For eg. 9 x 2 = 18 = 0 in R = Z18, 6 x 3 = 18 = 0 in R = Z18 and so on...

R = Z18 is not an integral domain because it has a lot many non zero zero divisors like 9,2,3,6,....(An integral domain is free from zero divisors ! )

Characteristic of a ring R is the least value or r such that ra = 0 for all a in R . Thus clearly, characteristic of Z18 = 18, for 18.a = 0 for all a in Z18.

It is a known fact that every maximal ideal is a prime ideal. But the converse holds only in finite commutative ring with unity. Our ring R = Z18 is finite commutative ring with unity.Hence every prime ideal is maximal as well.

The ideals <2> = {0,2,4,6,8,10,12,14,16} and <3> = {0,3,6,9,12,15} are both prime and maximal ideals in Z18, whereas <9> = {0,9} is not prime and hence not maximal since 9 = 3 x 3 but 3 is not an element of <9> .Similarly <6> = {0,6,12} is also not prime/maximal since 6 = 2.3 but both 2 and 3 are not elements of <6>.

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