Definite article required or not?

I think either a or the is required in the context. If you do not use an article , the sentence becomes ungrammatical

An Indicator of a strong character is a / the person' s ability to compliment the achievements of others.

I thiink it can be simplified as:

An indicator of a strong character is a/ the person's ability to compliment others' achievements.

If you want to talk about persons in general you have to use the article ** a**. or if you want to talk about a particular person you have to use the article ** the**

I think the use of definite article restricts the general sense.and I tend to use a in the sentence

Certainly an article of some sort is required here, and I personally prefer a to the. But you can easily avoid the problem:

An indicator of a strong character is the ability to compliment the achievements of others.

It is required.

In English, we have what are called noun phrases, and as described by this source:

In normal writing, nouns nearly always feature in noun phrases. It is rare to find a noun functioning by itself (i.e., without any modifiers) in a sentence.

Nouns are rarely used by themselves in noun phrases, a notable exception being when we use them as subjects when talking about the attributes of categories:

Boys are great!
Life is tough.
Drugs are a problem.