Die Persönlichkeit der Frau ist ruhig.

The subject is Die Persönlichkeit der Frau.

The main noun in that subject is Persönlichkeit (personality), and der Frau means of the woman. Because Persönlichkeit is singular, the verb is ist.

So the structure is:

• Die Persönlichkeit der Frau = the woman's personality
• ist = is
• ruhig = calm / quiet

Because der here is not masculine nominative der. It is the genitive singular form used with feminine nouns.

For die Frau, the article changes like this:

• nominative: die Frau
• accusative: die Frau
• dative: der Frau
• genitive: der Frau

So der Frau here means of the woman.

You know from the meaning and the structure.

Here, der Frau shows possession or relationship: the personality of the woman. That is a classic use of the genitive.

There is no preposition or verb here that would require the dative. So in this sentence, der Frau is understood as genitive.

German usually shows possession with the genitive case, not with an apostrophe.

So English:

• the woman’s personality

German:

• die Persönlichkeit der Frau

German does sometimes use something closer to an apostrophe-like possession with names, for example Marias Buch, but with normal nouns like die Frau, the genitive structure is standard.

Because ruhig comes after ist and works as a predicate adjective. Predicate adjectives in German do not take adjective endings.

Compare:

• Die Persönlichkeit ist ruhig.
  Here ruhig is after ist, so it has no ending.

• eine ruhige Persönlichkeit
  Here ruhige comes before the noun, so it is an attributive adjective and does take an ending.

This is a very common pattern in German:

• Der Mann ist alt.
• eine alte Frau
• Das Haus ist groß.
• ein großes Haus

It is grammatical, but it sounds a little formal or abstract.

A native speaker would often say one of these instead:

• Die Frau ist ruhig.
• Sie hat eine ruhige Persönlichkeit.

Die Persönlichkeit der Frau ist ruhig is understandable, but it sounds more analytical, like something from a description, report, or psychological discussion.

So the sentence is correct, but not the most everyday way to express the idea.

Because this is a normal main clause, and in German main clauses the finite verb usually comes in second position.

Here the first element is the subject:

• Die Persönlichkeit der Frau = position 1

Then comes the verb:

• ist = position 2

Then the complement:

• ruhig

So the pattern is:

• Subject + verb + complement

This is one of the most basic German sentence patterns.

It is an adjective.

In this sentence, it describes the subject through the verb sein (to be). That makes it a predicate adjective.

So:

• ruhig is not an adverb here
• it is an adjective describing die Persönlichkeit

Its meaning in this kind of sentence is usually closer to:

• calm
• composed
• quiet in nature

not just silent

Because all nouns are capitalized in German.

So:

• die Persönlichkeit
• die Frau

This is a basic German spelling rule. Adjectives like ruhig are normally not capitalized unless they begin the sentence or are used as nouns.

You could say it, and people would understand you, but it sounds less good here.

For simple possession, German usually prefers the genitive in writing and in more careful speech:

• die Persönlichkeit der Frau

Using von is often more common in everyday speech with some nouns, but in this sentence it sounds a bit clumsy:

• die Persönlichkeit von der Frau

A much more natural alternative would be:

• Die Frau hat eine ruhige Persönlichkeit.

Here it most naturally means calm.

For a personality, ruhig usually describes someone as:

• calm
• steady
• not excitable
• peaceful in manner

It does not necessarily mean the person is physically silent at that moment. So in this sentence, ruhig is about temperament, not just noise.

Because Persönlichkeit is a feminine noun in German: die Persönlichkeit.

It is also the subject here, so it is in the nominative case. The feminine nominative singular definite article is die.

So:

• die Persönlichkeit = the personality

If you learn nouns together with their articles, this kind of sentence becomes much easier to understand.