who are the greatest songwriter's of the 1900's?  Aside from Gershwin.

Irving Berlin
Cole Porter
Rodgers & Hart
Rodgers & Hammerstein
De Sylva, Brown & Henderson
Jerome Kern
Cy Coleman & Dorothy Fields
Kandor & Ebb

How many do you want?

I find it hard to believe that George Gershwin was a "great song
writer" in the 1900s. In 1900 he was 2 years old, and at the end of
that decade, in 1910, he was 12.

Leiber & Stoller. Definitely.

Poli-guru is correct to read "1900s" as the first decade of the
century, but since we speak and write of the mid and late nineteen
hundreds, I think most people accept it as short hand for the
twentieth century.

Is this all being left to me?

Here are some more:

Frank Loesser
Johnny Mercer
Gus Kahn
Jules Styne
Harry Warren
Al Dubin
Jay Livingstone
Ray Evans
Ira Gershwin (I assume that the question referred to George)

Oh yes and I have also assumed that the question means songwriters who
were active between 1 Jan 1900 and 31 Dec 1999.