To determine the nilpotent and idempotent elements in the ring [tex]\(\mathbb{Z}_{10}\)[/tex], we need to delve into the definitions and properties of these elements within the modular arithmetic context. Let's go through each definition and find the appropriate elements:
### Nilpotent Elements
An element [tex]\(x\)[/tex] in a ring is called nilpotent if there exists some positive integer [tex]\(n\)[/tex] such that [tex]\(x^n \equiv 0 \mod 10\)[/tex].
For [tex]\(\mathbb{Z}_{10} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\)[/tex], we systematically check the power of each element [tex]\(x\)[/tex] until we determine if [tex]\(x^n \equiv 0 \mod 10\)[/tex].
- [tex]\(0\)[/tex] to any power is still [tex]\(0\)[/tex], so [tex]\(0\)[/tex] is nilpotent.
- Any other number like [tex]\(1, 2, 3, \ldots, 9\)[/tex] raised to any power will never yield a result congruent to [tex]\(0 \mod 10\)[/tex].
Therefore, the only nilpotent element in [tex]\(\mathbb{Z}_{10}\)[/tex] is [tex]\(0\)[/tex].
### Idempotent Elements
An element [tex]\(x\)[/tex] in a ring is called idempotent if [tex]\(x^2 \equiv x \mod 10\)[/tex].
To find the idempotent elements in [tex]\(\mathbb{Z}_{10}\)[/tex], we test each element [tex]\(x\)[/tex] by calculating [tex]\(x^2\)[/tex] and checking if it is congruent to [tex]\(x \mod 10\)[/tex].
Performing these checks:
- [tex]\(0^2 \equiv 0 \mod 10\)[/tex]
- [tex]\(1^2 \equiv 1 \mod 10\)[/tex]
- [tex]\(2^2 \equiv 4 \not\equiv 2 \mod 10\)[/tex]
- [tex]\(3^2 \equiv 9 \not\equiv 3 \mod 10\)[/tex]
- [tex]\(4^2 \equiv 16 \not\equiv 4 \mod 10\)[/tex]
- [tex]\(5^2 \equiv 25 \equiv 5 \mod 10\)[/tex]
- [tex]\(6^2 \equiv 36 \equiv 6 \mod 10\)[/tex]
- [tex]\(7^2 \equiv 49 \not\equiv 7 \mod 10\)[/tex]
- [tex]\(8^2 \equiv 64 \not\equiv 8 \mod 10\)[/tex]
- [tex]\(9^2 \equiv 81 \not\equiv 9 \mod 10\)[/tex]
Thus, the idempotent elements in [tex]\(\mathbb{Z}_{10}\)[/tex] are [tex]\(0, 1, 5,\)[/tex] and [tex]\(6\)[/tex].
### Summary
- Nilpotent elements in [tex]\(\mathbb{Z}_{10}\)[/tex]: [tex]\(\{0\}\)[/tex]
- Idempotent elements in [tex]\(\mathbb{Z}_{10}\)[/tex]: [tex]\(\{0, 1, 5, 6\}\)[/tex]
These are the sets of nilpotent and idempotent elements in [tex]\(\mathbb{Z}_{10}\)[/tex].
