Consider the rings
๐ฎ1 = โค[๐ฅ] / ⟨2, ๐ฅ3⟩ ⁄ and ๐ฎ2 = โค2[๐ฅ] / ⟨๐ฅ2⟩
where ⟨2, ๐ฅ3⟩ denotes the ideal generated by {2, ๐ฅ3} in โค[๐ฅ] and ⟨๐ฅ2⟩ denotes the ideal generated by ๐ฅ2ย in โค2[๐ฅ].
Which of the following statements is/are correct?ย
1
Every prime ideal of ๐ฎ1 is a maximal idealย
2
๐ฎ2 has exactly one maximal idealย
3
Every element of ๐ฎ1 is either nilpotent or a unit
4
There exists an element in ๐ฎ2 which is NEITHER nilpotent NOR a unit