I am seeking a full time PhD admission.
I am seeking for admission to the full time PhD.
Both sentence seem to me correct. Am I right?
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We can use:
I am seeking admission to the full-time PhD program.
I seek admission to the full-time PhD program.
The second option is more formal, and therefore more appropriate when writing an objective, as you said you were doing in your comment. (I assume you are writing this for a formal or academic purpose.)
While we most often use the continuous to describe a present temporary action,
In special circumstances, we can also use the present simple to describe temporary states and actions. . . .
[For example] wants and likes want, like, need, hate, prefer
(Parrott, Martin. 2000. Grammar for English Language Teachers. Cambridge University Press, p. 155.)
Just as we would say,
I want to apply to the full-time Ph.D. program.
Or, I want a cup of tea.
Ph.D. program (with periods) is also acceptable. I would follow whichever practice the school does.