Try:

SELECT C.CustomerId, C.CustomerName, C.StatusId, I.AuthorityId
FROM Customer C
LEFT JOIN Identifier I ON I.CustomerId = C.CustomerId and I.AuthorityId = 11
WHERE C.StatusId = 1
ORDER BY C.CustomerName

If you want to add a and b and either may be null, you could use coalesce, which returns the first non-null parameter you pass it:

coalesce(a+b, a, b)

So in this case, if neither parameter is null, it will return the sum. If only b is null, it will skip a+b and return a. If a is null, it will skip a+b and a and return b, which will only be null if they are both null.

If you want the answer to be 0 rather than null if both a and b are null, you can pass 0 as the last parameter:

coalesce(a+b, a, b, 0)

Do consider @erwins answer - null might not be the right thing to be using.

As per the comments, I'm going to assume you're getting mixed up between update and insert.

 USE [DB_A] ;
 UPDATE dbo.address
 SET City = 'test',
     State = 'ing'
 WHERE City IS NULL ;

And

 USE [DB_A] ;
 UPDATE dbo.address
 SET City = NULL,
     State = NULL
 WHERE City = 'test' ;

I think you're there otherwise :-)

INSERT INTO atable (x,y,z) VALUES ( NULL,NULL,NULL)

I think Zack properly answered the question but just to cover all the bases:

Update myTable set MyColumn = NULL

This would set the entire column to null as the Question Title asks.

To set a specific row on a specific column to null use:

Update myTable set MyColumn = NULL where Field = Condition.

This would set a specific cell to null as the inner question asks.

They are different. When you enclose something in single quotes, it becomes a string -- this is true of all keywords in SQL, including NULL.

When you use just NULL, it is the SQL keyword, and it is compatible with all types.

So, these are quite different:

insert into t (col1, col2)
    values ('2021-05-25 11:58:41.000', NULL);

and:

insert into t (col1, col2)
    values ('2021-05-25 11:58:41.000', 'NULL');

'NULL' is a string and only compatible with a string type. You should get a type-conversion error if the column is not a string.

The one caveat is if you are storing the values in a text file for loading into the database. In such a file, strings may or may not be delimited with single quotes, depending on the structure of the file.