"One-way synchronization" between Mail and Gmail?
You are describing an account set up with POP3.
To achieve that behavior follow this instructions:

First, set up POP in Gmail


On your computer, open Gmail.
In the top right, click Settings Settings.
Click Settings.
Click the Forwarding and POP/IMAP tab.
In the "POP Download" section, select Enable POP for all mail or Enable POP for mail that arrives from now on.
At the bottom of the page, click Save Changes.
Next, make changes on your email client
Go to your client, like Microsoft Outlook, and check these settings.


Incoming Mail (POP) Server: pop.gmail.com

Requires SSL: Yes

Port: 995

Outgoing Mail (SMTP) Server: smtp.gmail.com

Requires SSL: Yes

Requires TLS: Yes (if available)

Requires Authentication: Yes

Port for SSL: 465

Port for TLS/STARTTLS: 587

If you use Gmail with your work or school account, enter mail.domain.com, then select Port 110.

Server timeouts Greater than 1 minute (5 is recommended)
Full Name or Display Name   Your name
Account Name, User Name, or Email address   Your email address
Password    Your Gmail password

When adding your account choose "Other" instead of "Gmail" on Mail app.