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    I have a theorem which says:

    If the ideal  I is maximal, then  R/I is a field.

    Is the converse true, can you say that if  R/I is a field, then  I is maximal??

    Thanks!
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    (Original post by adie_raz)
    I have a theorem which says:

    If the ideal  I is maximal, then  R/I is a field.

    Is the converse true, can you say that if  R/I is a field, then  I is maximal??

    Thanks!
    You need R to be commutative as well don't you?

    The converse is true. Perhaps you have a theorem that says:

    If R is commutative with an identity element then R is a field if and only if the only ideals of R are {0} and R

    Also, you need the third isomorphism theorem:

    If R is a ring and I and ideal of R, there is a one to one correspondence between ideals J so that I \subseteq J \subseteq R and ideals of R/I.

    Then R/I is a field if and only if the ideals of R/I are R/I and {0} if and only if there are no ideals strictly between I and R (which is what it means for I to be maximal).
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    (Original post by SsEe)
    You need R to be commutative as well don't you?

    The converse is true. Perhaps you have a theorem that says:

    If R is commutative with an identity element then R is a field if and only if the only ideals of R are {0} and R

    Also, you need the third isomorphism theorem:

    If R is a ring and I and ideal of R, there is a one to one correspondence between ideals J so that I \subseteq J \subseteq R and ideals of R/I.

    Then R/I is a field if and only if the ideals of R/I are R/I and {0} if and only if there are no ideals strictly between I and R (which is what it means for I to be maximal).
    Thanks for the reply! yes I am working with commutative rings (sorry I didn't mention that).

    Thank you
 
 
 
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