micromass said:This statement is 100% correct. So don't worry about it.
Have you heard of the fourth isomorphism theorem (you probably didn't call it that). It states that there is a bijective correspondance between ideals of R that contain I and ideals of R/I.
In particular, if J is an ideal of R/I, then J=J'/I for some ideal J' of R.
This is the thing you have to use to prove this question. If you didn't see it, then perhaps you could try to prove it...
A maximal ideal is a proper ideal in a ring that is not contained in any other proper ideal. In other words, it is an ideal that cannot be extended any further without becoming the entire ring.
A quotient ring, also known as a factor ring, is a ring constructed by taking a ring and "quotienting out" an ideal. This means that the elements of the ideal are considered to be equivalent to 0 in the quotient ring.
Maximal ideals play an important role in quotient rings because they are the largest possible ideals that can be quotiented out. In fact, a quotient ring is a field if and only if its only ideals are the zero ideal and the maximal ideal.
Maximal ideals are significant in abstract algebra because they can be used to define important concepts such as prime ideals and irreducible elements. They also have applications in algebraic geometry and commutative algebra.
To determine if a given ideal is maximal in a quotient ring, you can use the third isomorphism theorem. This theorem states that if I is an ideal of a ring R and J is an ideal of R that contains I, then there exists a unique ideal K of R/I such that J = I + K and J/I is isomorphic to K. If the given ideal is not contained in any other ideal, then it is maximal.